On the divergence of Birkhoff Normal Forms

Raphaël Krikorian

doi:10.1007/s10240-022-00130-2

On the divergence of Birkhoff Normal Forms

Krikorian, Raphaël 2022-06-01 00:00:00 by RAPHAËL KRIKORIAN To the memory of my father Grégoire Krikorian (1934–2018) ABSTRACT It is well known that a real analytic symplectic diffeomorphism of the 2d -dimensional disk (d ≥ 1) admitting the origin as a non-resonant elliptic ﬁxed point can be formally conjugated to its Birkhoff Normal Form, a formal power series deﬁning a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Pérez-Marco’s theorem (Ann. Math. (2) 157:557–574, 2003) is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when d = 1 the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus. CONTENTS 1. Introduction ...................................................... 2 2. Notations, preliminaries ............................................... 26 3. A no-screening criterion on domains with holes .................................. 33 4. Symplectic diffeomorphisms on holed domains .................................. 38 5. Cohomological equations and conjugations ..................................... 46 6. Birkhoff Normal Forms ................................................ 54 7. KAM Normal Forms ................................................. 60 8. Hamilton-Jacobi Normal Form and the Extension Property ............................ 68 9. Comparison Principle for Normal Forms . ..................................... 84 10. Adapted Normal Forms: ω Diophantine . ..................................... 88 11. Adapted Normal Forms: ω Liouvillian (CC case) ................................. 97 12. Estimates on the measure of the set of KAM circles . ............................... 103 13. Convergent BNF implies small holes ........................................ 109 14. Proof of Theorems C, A and A’ ........................................... 111 15. Creating hyperbolic periodic points ......................................... 113 16. Divergent BNF: proof of Theorems E, B and B’ .................................. 122 Acknowledgements ..................................................... 129 Appendix A: Estimates on composition and inversion .................................. 130 Appendix B: Whitney type extensions .......................................... 135 Appendix C: Illustration of the screening effect . .................................... 138 Appendix D: First integrals of integrable ﬂows ...................................... 139 Appendix E: (Formal) Birkhoff Normal Forms . ..................................... 141 Appendix F: Approximate Birkhoff Normal Forms ................................... 147 Appendix G: Resonant Normal Forms .......................................... 155 Appendix H: Approximations by vector ﬁelds . ..................................... 161 Appendix I: Adapted KAM domains: lemmas . ..................................... 167 Appendix J: Classical KAM measure estimates . . .................................... 170 Appendix K: From (CC) to (AA) coordinates ....................................... 171 Appendix L: Lemmas for Hamilton-Jacobi Normal Forms ............................... 173 Appendix M: Some other lemmas . ........................................... 175 Appendix N: Stable and unstable Manifolds ....................................... 177 References ......................................................... 179 This work was supported by a Chaire d’Excellence LABEX MME-DII, the project ANR BEKAM: ANR-15- CE40-0001 and an AAP project from CY Cergy Paris Université. © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2022 https://doi.org/10.1007/s10240-022-00130-2 2 RAPHAËL KRIKORIAN 1. Introduction We consider in this paper real analytic diffeomorphisms deﬁned on an open set of the d d d d 2d -cartesian space R × R or respectively of the 2d -cylinder (or annulus) (R/2π Z) × R (d ≥ 1), which are symplectic with respect to the canonical symplectic forms dx ∧ dy , j j j=1 d d d d (x, y) ∈ R × R , resp. dθ ∧ dr , (θ , r) ∈ (R/2π Z) × R , and leave invariant j j j=1 d d d d d {(0, 0)}∈ R × R ,resp.the torus T := (R/2π Z) ×{0}⊂ (R/2π Z) × R .Weshall d d d assume that the invariant sets {(0, 0)}∈ R × R , resp. (R/2π Z) ×{0},are elliptic equilib- rium sets in the following sense: there exists ω = (ω ,...ω ) ∈ R ,the frequency vector,such 1 d that d d 2 f : (R × R ,(0, 0)) ý, f = Df (0, 0) ◦ (id + O (x, y)) (1.1) ±2π −1ω spec(Df (0, 0))={e , 1 ≤ j ≤ d} and respectively d d 2 (1.2) f : ((R/2π Z) × R , T ) ý, f (θ , r) = (θ + 2πω, r) + (O(r), O(r )). If ω = ω for i = j (stronger non-resonance condition will be made later), the deriva- i j tive Df (0, 0) of f at the ﬁxed point (0, 0) in (1.1) can be symplectically conjugated to a symplectic rotation and we can thus assume Df (0, 0) is a symplectic rotation: for any x = (x ,..., x ), y = (y ,..., y ), x = ( x ,..., x ), y = ( y ,..., y ) one has (i = −1) 1 d 1 d 1 d 1 d 2π iω x + i y = e (x + iy ) j j j j Df (0, 0) · (x, y) = ( x, y) ⇐⇒ ∀ 1 ≤ j ≤ d. We shall refer to situation (1.1)asthe Elliptic ﬁxed point or the Cartesian Coordinates ((CC) for short) case and to situation (1.2)asthe Action-Angle ((AA) for short) case. Important examples of such diffeomorphisms are provided by ﬂows ( ) ,orby t∈R suitable Poincaré sections on some energy level, of Hamiltonian systems ∂ H ∂ H ∂ H ∂ H x˙ = (x, y), y˙ =− (x, y), resp. θ = (θ , r), r˙=− (θ , r) ∂ y ∂ x ∂ r ∂θ d d d d where H : (R × R ,(0, 0)) → R resp. H : ((R/2π Z) × R , T ) → R (d = d or d = d + 1) is real analytic and satisﬁes 2 2 x + y j j (1.3) (CC)-case H(x, y) = 2π ω + O (x, y), j=1 (1.4) (AA)-case H(θ , r) = 2π ω r + O(r ). j j j=1 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 3 If we denote by the time-1 map of a Hamiltonian H and deﬁne the observable 2 2 r : (x, y) → (1/2)(x + y ), resp. r : r → r (1 ≤ j ≤ d ) we can write (1.1), resp. (1.2), as j j j j j d d 2 (1.5) (CC)-case f : (R × R ,(0, 0)) ý, f = + O (x, y) 2πω,r d d 2 (1.6) (AA)-case f : ((R/2π Z) × R , T ) ý, f = + (O(r), O(r )) 0 2πω,r where ω, r= ω r , r = (r ,..., r ). j j 1 d j=1 The representations (1.5), resp. (1.6), give a very rough understanding of the be- havior of the ﬁnite time dynamics of the diffeomorphism f in a neighborhood of the elliptic equilibrium sets {(0, 0)}, resp. T : it is interpolated by the dynamics of which is 0 2πω,r quasi-periodic in the sense that all its orbits are quasi-periodic with frequencies ω ,...,ω . 1 d Improving this approximation is an old and important problem (it was a central theme of research of the astronomers of the XIXth century; see the references of the very in- structive introduction by Pérez-Marco in [36]) that has a solution at least in the (CC)-case (1.5) if the frequency vector ω is non-resonant:any relation k + k ω +···+ k ω = 0with 0 1 1 d d k , k ,..., k ∈ Z implies that k = k =··· = k = 0. Indeed, after using nice changes of 0 1 d 0 1 d coordinates (symplectic transformations) one can interpolate, in small neighborhoods of the origin, the dynamics of f by quasi-periodic ones with much better orders of approxima- tion and for much longer times. There are two remarkable features of this interpolation: the ﬁrst, is that the frequencies of the interpolating quasi-periodic motions now depend on the initial point and do not necessarily coincide with the frequencies at the origin; the second, is that if we push the order of approximation, these frequencies stabilize in a way. This is the content of the famous Birkhoff Normal Form Theorem, formalized by Birkhoff in the 1920’s [5], [4], [49] and which paved the way to the major achievements of the KAM theory (named after Kolmogorov, Arnold and Moser) in the 1960’s, on the existence of (inﬁnite time) quasi-periodic motions for a wide class of diffeomorphisms of the form (1.1), (1.2); see [29], [1], [32](and[34] for ﬁnite time approximations). We now describe in more details the Birkhoff Normal Form Theorem. 1.1. Birkhoff Normal Forms. — From now on, we assume that ω is non-resonant. We begin with the Elliptic ﬁxed point case ((CC)-case). The ﬁrst statement of the Birkhoff Normal Form Theorem is the following. For any N ∈ N , there exist a poly- nomial B ∈ R[r ,..., r ],B (r) = 2πω, r+ O(r ), of total degree N and a symplec- N 1 d N d d tic diffeomorphism Z : (R × R ,(0, 0)) ý (preserving the standard symplectic form dx ∧ dy and tangent to the identity Z = id + O (x, y))suchthat k k N k=1 −1 2N+1 (1.7)Z ◦ f ◦ Z (x, y) = (x, y) + O (x, y). N B N N 1 −α k k For (x, y)ε-close to (0, 0) and n ∈ N not too large n = O(ε ),0 <α < 1 the iterates f (x, y), k ≤ n (f denotes 2−α k the composition f ◦···◦f , k times) stay ε -close to those of the symplectic rotation, (x, y). 2πω,r 4 RAPHAËL KRIKORIAN d d The diffeomorphism : (R × R , 0) ý is a generalized symplectic rotation i∂ B (r) j N x + i y = e (x + iy ) j j j j (1.8) (x, y) = ( x, y) ⇐⇒ ∀ 1 ≤ j ≤ d 2 2 2 2 (recall r = ((1/2)(x + y ), ...,(1/2)(x + y )))and deﬁnes an integrable dynamics in a 1 1 d d strong sense: every orbit of is quasi-periodic and, in addition, the origin is Lyapunov ∗ d stable. Indeed, for each c = (c ,..., c ) ∈ (R ) ,the d -dimensional torus 1 d 2d 2 2 T := {(x, y) ∈ R , ∀ 1 ≤ j ≤ d, r := (1/2)(x + y ) = c } c j j j j is globally invariant by and the restricted dynamics of on the torus T T := B B c N N d d d d R /(2π Z) is conjugated to a translation T θ → θ + 2πω(c) ∈ T with frequency vector −1 ω(c) = (2π) ∇ B (c). The dynamics of is thus completely understood on the whole N B 2 d d phase space R × R . Here comes the second part of the statement. The polynomials B and the com- ponents of Z − id converge as formal power series when N goes to inﬁnity: B → B ∈ N N ∞ R[[r ,..., r ]],Z → Z ∈ R[[x, y]] and, in the set of formal power series R[[x, y]],one 1 d N ∞ has the following formal conjugacy relation −1 (1.9)Z ◦ f ◦ Z (x, y) = (x, y). ∞ B ∞ ∞ The formal power series B is unique if Z is tangent to the identity and is therefore ∞ ∞ invariant by (smooth or formal) conjugations tangent to the identity; it is called the Birkhoff Normal Form (BNF for short) of f and we shall denote it by BNF(f ): BNF(f ) = B (r ,..., r ) ∈ R[[r ,..., r ]]. ∞ 1 d 1 d On the other hand the formal conjugacy Z , which is called the normalization transforma- tion, is not unique (but if properly normalized is unique). The preceding results hold in the Action-Angle case (1.6) but under a Diophantine assumption on ω (this is stronger than mere non-resonance): (1.10) ∀ k ∈ Z {0}, min|k,ω− l|≥ (τ ≥ d). l∈Z |k| The positive numbers τ and κ are called respectively the exponent and the constant of the 3 ∗ Diophantine condition. One can then prove similarly the existence: (a) for any N ∈ N , When c has some zero components, T is a d -dimensional torus, 0 ≤ d ≤ d , and the restricted dynamics of c c c B on T is again conjugate to a translation on a torus. 3 d The set of vectors of R satisfying a Diophantine condition with ﬁxed exponent τ and ﬁxed constant κ has positive Lebesgue measure if τ> d and if κ> 0 is small enough; for each τ> d , the union of these sets on all κ> 0has full Lebesgue measure in R . ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 5 of a polynomial B ∈ R[r ,..., r ],B (r) = 2πω, r+ O(r ) and of a symplectic dif- N 1 d N d d feomorphism Z : ((R/2π Z) × R , T ) ý (preserving the standard symplectic form N 0 dθ ∧ dr ,Z = id + (O(r), O(r )))suchthat k k N k=1 −1 N N+1 (1.11) Z ◦ f ◦ Z (θ , r) = (θ , r) + (O (r), O (r)) N B (1.12) (θ , r) = (θ +∇ B (r), r) B N ( is called an integrable twist); and: (b) of a formal power series B ∈ R[[r ,..., r ]], B ∞ 1 d the Birkhoff Normal Form, and of a formal symplectic transformation Z = id + 2 ω d (O(r), O(r )) in C (T )[[r ,..., r ]] (the set of formal power series with coefﬁcients in 1 d d ω d the set of real analytic functions T → T) such that one has in C (T )[[r ,..., r ]] the 1 d formal conjugation relation −1 (1.13)Z ◦ f ◦ Z (θ , r) = (θ +∇ B (r), r). ∞ ∞ Again we denote BNF(f ) = B (r ,..., r ) ∈ R[[r ,..., r ]]. ∞ 1 d 1 d All the preceding discussion on Birkhoff Normal Forms holds if we only assume f to be smooth. We can summarize this: d d Theorem (Birkhoff).— Any smooth symplectic diffeomorphism f : (R × R ,(0, 0)) ý d d (d ≥ 1)(resp. f : ((R/2π Z) × R , T ) ý) admitting the origin as a non-resonant elliptic ﬁxed point (resp. of the form (1.6) with ω Diophantine) is formally (strongly) integrable: it is conjugated in ∞ d R[[x, y]] (resp. C ((R/2π Z) )[[r]]) to the formal generalized symplectic rotation (resp. the formal integrable twist) . The formal series BNF(f ) is an invariant of formal conjugation. BNF(f ) We refer to [4]and [49] (Section 24) for a proof of the preceding theorem in the case of symplectic diffeomorphisms of the disk admitting a non-resonant elliptic ﬁxed point and to [13] for the case of Hamiltonian systems admitting a Diophantine KAM torus. We shall reformulate (in the real analytic case) this theorem in Section 6, cf. Propo- sitions 6.1–6.2, and shall give a proof of it in Section E of the Appendix where we mainly concentrate on the AA-case. These formal (and approximate) Birkhoff Normal Forms can be deﬁned in the ∂ H ∂ H more classical setting of Hamiltonian ﬂows x˙ = (x, y), y˙ =− (x, y) (or θ = ∂ y ∂ x ∂ H ∂ H t (θ , r), r˙ =− (θ , r)): f and (B = BNF(f )) are then replaced by ( ) and B t∈R ∂ r ∂θ t 5 ( ) in (1.9), (1.13)(we shallthenwrite B = BNF(H) ). t∈R In the Hamiltonian case, there is a weaker notion of integrability, usually called Poisson integrability, which corresponds to the situation where the considered Hamiltonian has a complete system of functionally independent integrals (observables constant under the motion) which commute for the Poisson bracket. Poincaré discovered [37] that, in 4 k In the C category, one can deﬁne B and Z up to some order N depending on k but one cannot deﬁne in N N general BNF(f ). A more classic equivalent formulation is H = B ◦ Z. 6 RAPHAËL KRIKORIAN general, real analytic Hamiltonian ﬂows do not admit other analytic ﬁrst integrals than the Hamiltonian itself and hence that in general no relation like (1.9) can hold with con- verging Z and B .Siegelproved[48] in 1954 (see also [47], [49], [52], [36]) that, ∞ ∞ whatever the ﬁxed non-resonant frequency vector at the origin ω is, the normalizing con- jugation Z cannot in general deﬁne a convergent series. Indeed, the existence of a convergent normalizing transformation yields real analytic Poisson integrability afact (known to Birkhoff [5]) that is not compatible with the richness of a generic dynamics near a non-resonant elliptic equilibrium. Note that the converse statement is true: real analytic Poisson integrability implies the existence of a real analytic normalizing Birkhoff transformation (cf. [25], [28], [56]). As for the Birkhoff Normal Form itself, H. Eliasson formulated the following natu- ral question [11], [10] (see also the references in [36]): Question A (Eliasson).— Are there examples of real analytic symplectic diffeomorphisms or Hamiltonians admitting divergent (i.e. with a null radius of convergence) Birkhoff Normal Form? The preceding question has an easy positive answer in the smooth case (the map f is only assumed to be smooth): indeed, one can choose f to be of the form f = where : (R , 0) → R is smooth with a divergent Taylor series at the origin; since equalities (1.9)(1.13) only depend on the inﬁnite jet J(f ) of f at 0, the special integrable form of f implies BNF(f ) = J(f ) thus BNF(f ) is diverging. The situation is not so clear if f is real analytic. In contrast with the aforementioned generic divergence of the normalizing 9 10 transformation, there seems to be a priori no obvious dynamical obstruction to the divergence of the Birkhoff Normal Form. The ﬁrst breakthrough in connection with Eliasson’s question came from R. Pérez- Marco [36] who proved, in the setting of Hamiltonian systems having a non-resonant elliptic ﬁxed point, the following dichotomy: Theorem (Pérez-Marco [36]).— For any ﬁxed non-resonant frequency vector ω ∈ R , d ≥ 2, one has the following dichotomy: either for all real analytic Hamiltonian H of the form (1.3) BNF(H) converges (deﬁnes a converging analytic series) or there is a “prevalent” set of such H for which BNF(H) diverges. We refer to Section 1.4 for a precise deﬁnition of “prevalent”. A similar dichotomy holds in the setting of real analytic symplectic diffeomorphisms in the (CC)-case, and, Here it means G -dense in some set of real analytic functions with ﬁxed radius of convergence. This phenomenon is even “prevalent” as shown by Pérez-Marco [36]. If Z converges the observables r ◦ Z , j = 1,... , d are a complete set of real analytic and functionally indepen- ∞ j ∞ dent Poisson commuting integrals. By which we mean the coexistence of quasi-periodic motions and hyperbolic behavior in any neighborhood of the equilibrium; see for a global view on these topics and references the book [2]. We shall in fact see in this paper that there are such dynamical obstructions. Like the accumulation at the origin of hyperbolic periodic points or normally hyperbolic tori. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 7 both in the Hamiltonian or diffeomorphism framework, it can be extended to the (AA)- case (but under the stronger assumption that ω is Diophantine); cf. Theorem 1.3 of our paper. Pérez-Marco’s argument is not based on an analysis of the dynamics of f but rather focuses on the coefﬁcients of the BNF and exploits their polynomial dependence on the coefﬁcients of the initial perturbation by using techniques from potential theory. The following two Theorems are an answer (in the symplectomorphism setting) to Eliasson’s question and decide which of the two assertions of Pérez-Marco’s alternative holds (see Theorem D of Section 1.4 for a more precise statement). Main Theorem 1 ((CC)-Case).— For any d ≥ 1 and any non-resonant frequency vector d d d ω ∈ R , there exists a “prevalent” set of real analytic symplectic diffeomorphism f : (R × R , (0, 0)) ý of the form (1.5) the Birkhoff Normal Forms of which are divergent. In the Action-Angle Case (1.6) it takes the following form: Main Theorem 1’ ((AA)-Case).— For any d ≥ 1 and any Diophantine frequency vector d d ω ∈ R , there exists a “prevalent” set of real analytic symplectic diffeomorphism f : ((R/2π Z) × R , T ) of the form (1.6) the Birkhoff Normal Forms of which are divergent. Main Theorems 1, 1’ also extend to the Hamiltonian case (1.3)–(1.4) (with d = d + 1). Note that from Pérez-Marco’s Theorem (and its analogue in the symplectomor- phism case), in order to prove that the divergence of the Birkhoff Normal Form holds in a prevalent way, it is enough to provide, for each ﬁxed frequency vector ω, one example for which the BNF is divergent. On the other hand, if one is able to construct one such example for some d , then it is easy to construct other such examples for any d > d (see 0 0 for example the proof of Theorem D). Proving Main Theorem 1 (resp. 1’)thusamounts to constructing when d = 1, for each irrational (resp. Diophantine) ω ∈ R, one example of a real analytic symplectic diffeomorphism with a diverging BNF. Gong already provided in [17] (by a direct analysis of the coefﬁcients of the BNF) 2 2 3 examples of real analytic Hamiltonians ω, r+ F : (R × R , 0) → R,F = O (x, y),with Liouvillian frequency ω ∈ R at the origin and a divergent BNF and Yin [54] produced analogue of Gong’s examples in the diffeomorphism case (area preserving map of (R , 0) with a very Liouvillian elliptic ﬁxed point). In these examples the divergence of the BNF is caused by the presence of very small denominators (due to the Liouvillian character of ω) appearing in the coefﬁcients of the BNF. After our result was announced, Fayad [15]con- structed simple examples of real analytic Hamiltonian systems in (R , 0) (d = d + 1 = 4 The idea of using potential theory in problems of small denominators was ﬁrst introduced by Yu. Ilyashenko [23]. See [35] for further references. It is not clear whether one can, for general systems, deduce the case of Hamiltonian ﬂows from the case of diffeomorphisms and vice versa. On the other hand the proofs of Main Theorems 1, 1’ and in particular the proofs of Main Theorem 2 and of Theorems A–B, A’–B’ below extend to Hamiltonian ﬂows with 1 + 1 degrees of freedom. 8 RAPHAËL KRIKORIAN degrees of freedom) with any ﬁxed non-resonant frequency vector at the origin and diver- gent BNF. The argument again is based on an analysis of the coefﬁcients of the BNF; one considers Hamiltonians with two degrees of freedom where two extra action variables are added as formal parameters, one of them appearing later in the denominators of the BNF. These types of examples can be constructed in the diffeomorphism case for d ≥ 3. In a different context, that of reversible systems, let us mention a result of divergence of normal forms in [19] based on a different method (control of coefﬁcients growth) and a result of divergence of normalizing transformations in [33]. We now formulate Eliasson’s question in a stronger form: Question B. — Does the convergence of a formal conjugacy invariant like the Birkhoff Normal Form of a real analytic symplectic diffeomorphism (or Hamiltonian) have consequences on the dynamics of the diffeomorphism (or Hamiltonian)? Note that the convergence of the normalizing transformation has an obvious con- sequence, namely, integrability. As for Question B, there are various results pointing to some kind of rigidity phenomena if analyticity (and some arithmetic properties on ω)is assumed. To be more speciﬁc, let us mention a striking one: Bruno [7] and Rüssmann [42]provedthatif f is real analytic and if its BNF is trivial,BNF(f ) = 2πω, r (in par- ticular BNF(f ) converges), then f is real analytically conjugated to , provided the 2πω,r frequency vector at the origin ω satisﬁes a Diophantine condition. We refer to [53], [25], [11], [9], [51], [18], [13], [12] for generalizations of the Bruno-Rüssmann Theorem and related results. The Main Result of our paper is in some sense one answer, amongst possibly oth- ers, to the previous question at least when d = 1 and if f is assumed to satisfy some twist condition. Let us say that a diffeomorphism of the form (R , 0) ý (1.1)or (R × T, T ) ý is twist (or satisﬁes a twist condition) if the second order term of its BNF is not zero: −1 2 3 (2π) BNF(f )(r) = ωr + b r + O(r ), b = 0. 2 2 Main Theorem 2. — If the Birkhoff Normal Form of a real analytic symplectic twist diffeomor- phism (R , 0) ý (1.1)or (R × T, T ) ý (1.2) converges then the measure of the complement of the union of all invariant curves accumulating the origin is much smaller than what it is for a general such diffeomorphism. In other words, the convergence of a formal object like the BNF has consequences on the dynamics of the diffeomorphism. Precise statements are given in Theorems A–B, A’–B’)ofSection 1.2 and Theorems E and E’ of Section 1.4. Combined with (the exten- sion to the diffeomorphism case of) Pérez-Marco’s Theorem [36], this gives that in any number of degrees of freedom, a general real analytic symplectic diffeomorphism admit- An easily checkable condition. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 9 ting the origin as an non-resonant elliptic equilibrium has a divergent Birkhoff Normal Form (see Theorem D). Having in mind the aforementioned result by Bruno and Rüssmann, a natural stronger question is whether the following rigidity result is true: Question C. — Is it true that a real analytic symplectic diffeomorphism or Hamiltonian system having a Diophantine elliptic equilibrium and a non degenerate and convergent BNF is (real analytically) integrable (in some neighborhood of the origin)? The examples by Farré and Fayad in [14] of real analytic Hamiltonians on d+1 d+1 T × R with convergent BNF and with an unstable Diophantine elliptic torus show that such a generalization is not true for d ≥ 2, at least in the (AA) case and if by non de- generate we mean that BNF(f ) is not trivial. The question is still open for d < 2. Note that though in Farré-Fayad’s examples the BNF (which is explicit) is not degenerate (the rank of its quadratic part is not zero unlike in the Bruno-Rüssmann Theorem), its quadratic part is not of maximal rank. If one drops in Question C the Diophantine assumption and assumes the BNF to be trivial (like in Bruno-Rüssmann’s Theorem) the question is open (this question is related to a question of Birkhoff on pseudo-rotations and to the problem of constructing real analytic Anosov-Katok examples; cf. [16] for details and references). When d = 1 the situation might be more favorable. To any twist area preserving diffeomorphism f : (R , 0) ý (1.1)or f : (R × T, T ) ý (1.2) one can associate (we use the notations and terminology of [46]) its minimal action α : I → R (I is an open interval containing ω) that assigns to each ϕ ∈ I the average action of any minimal orbit with ro- tation number ϕ. The function α is strictly convex (in fact differentiable at any irrational) ∗ 16 and one can thus deﬁne its Legendre conjugate function α : r → sup (ϕr − α(ϕ)) ϕ∈I (see [30], [31], [46] for further details). The function r → α (r) (deﬁned on a neighbor- hood of 0) can be seen as a frequency map in the sense that if γ is an invariant circle for f with “symplectic height” (area with respect to the origin) c then α (c) is the rotation number of f restricted on γ . It has the following properties: the Taylor series of α at 0 coincides with the Birkhoff Normal Form of f ;moreover, if α (hence α) is differentiable 0 0 then f is C -integrable (see [46]). This C -integrability often yields rigidity (we refer to [3], [27] for an illustration of this fact in the context of billiard maps). The techniques de- veloped in our paper are probably enough to prove that if the function α is real analytic then f is in fact real analytically integrable. A more delicate issue is to establish real an- alyticity of α by only knowing that its Taylor series at 0 (the BNF) deﬁnes a converging series. Note that if f is real analytic one can construct dynamically relevant holomor- phic functions (frequency maps) deﬁned on complex domains having positive Lebesgue measure intersections (Cantor sets) with the real axis (see [8], [38]) and which coincide It seems that the (CC) case is not yet settled. Area preserving maps with no periodic points except the origin. 16 ∗ The functions α and α (also denoted β and α) are called Mather’s functions. 10 RAPHAËL KRIKORIAN on these intersections with α . The restrictions of these holomorphic functions on these Cantor sets have some quasi-analyticity properties but it seems that there are not strong enough to deduce that α behaves like a genuine quasi-analytic function (in particular that the convergence of the Taylor series at 0 implies analyticity); we refer to [8]for references and for more details. We conclude this subsection by the following question. Question D. — Is a given real analytic symplectic diffeomorphism accumulated by real analytic symplectic diffeomorphisms having convergent BNF’s? (We do not ask the radii of convergence of the BNF’s to be bounded below). Positive answers to Questions C, D would imply that any real analytic symplectic diffeomorphism admitting an elliptic equilibrium set is accumulated in the strong real analytic topology by diffeomorphisms of the same type that are in addition integrable in a neighborhood of the equilibrium set. 1.2. Invariant circles. — As suggest (1.7), (1.11) the BNF (more precisely its approx- imate version B ) is, as we have already mentioned, a precious tool to study the problem of the existence of quasi-periodic motions in the neighborhood of an elliptic equilibrium. A bright illustration of this fact is certainly the KAM Theorem ([29], [1], [32]) that yields, under suitable non-degeneracy conditions on the BNF (non-planarity), the existence of many KAM tori accumulating the origin (see [13], [12] for results under much weaker non- degeneracy assumptions). We shall be mainly concerned with the 2-dimensional case (d = 1) and we restrict to this case in this subsection. Recall our notation T = R/2π Z for the 1-dimensional torus. An invariant circle (or also invariant curve) for a real analytic (or smooth) diffeomor- phism f : (R × R,(0, 0)) ý of the form (1.5) is the image γ = g(T) of an injective 1 2 C map g : T → R {0} with index ±1at0such that f (γ ) = γ . Likewise, in the (AA) case, an invariant circle or invariant curve) for a real analytic (or smooth) diffeomor- phism f : (T × R, T ) ý of the form (1.6) is the image γ = g(T) of an injective C map g : T → T × R which is homotopic to the circle T = T×{0} and such that f (γ ) = γ . Note that in this latter case, by a theorem of Birkhoff (cf. [4], [21]), invariant circles close enough to T are in fact graphs if f satisﬁes a twist condition: −1 2 (1.14) b (f ) =0if (2π) BNF(f ) = ωr + b (f )r +··· . 2 2 This means that if the given diffeomorphism f has a holomorphic extension to some complex domain W there exists a slightly smaller subdomain W ⊂ W and a sequence of real-symmetric holomorphic diffeomorphisms f deﬁned on Wsuch thatlim sup |f − f |= 0. n→∞ n A KAM torus is an invariant Lagrangian torus on which the dynamics is conjugated to a linear translation with a Diophantine frequency vector. These curves are also called essential curves. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 11 In both cases, we denote by G the set of f -invariant curves and, for t > 0, by L (t) the f f set of points in M := R or T × R which belong to an invariant curve γ ∈ G such that R f γ ⊂ M ∩{|r| < t} L (t) = γ. γ∈G γ⊂M ∩{|r|<t} We then deﬁne (M = R × R or T × R) m (t) = Leb ((M ∩{|r| < t}) L (2t)). f M R f Notation 1.1. — We shall use the following notations: if a ≥ 0 and b > 0 are two real numbers we write a b for: “there exists a constant C > 0 independent of a and b such that a ≤ Cb”. If we want to insist on the fact that this constant C depends on a quantity β we write a b. We shall also write a b to say that a/b is small enough and a b to express the fact that this smallness condition depends on β . The notations b a, b a, b aand b a are deﬁned in the same way. When β β one has a band b a we write a b. The 2-dimensional version of the KAM Theorem is the celebrated Moser’s twist Theorem [32] (see also [43]): Theorem (Moser).— Let f be a symplectic smooth diffeomorphism like (1.5)or(1.6) satisfying the twist condition (1.14). If f admits Birkhoff Normal Forms at the origin to all orders then, for any constant a > 0, (1.15) m (t) t . Let us comment on the previous result. In the (CC) case, it is in fact enough to as- sume that ω in (1.5) is non-resonant since under this condition f admits Birkhoff Normal Forms to all orders. On the other hand, in the (AA) case the existence of the BNF to all orders, more precisely the existence of solutions to the related cohomological equations (see Lemma E.7), requires ω in (1.6) to be Diophantine; in the real analytic case a weaker arithmetic condition is enough, see [44], [45]. If in the (CC) case ω is non-resonant only up to some low order, (1.15) holds only for some a > 0 (see [38]). If in the (AA) case we drop the assumption that ω is Diophantine (but we assume ω to be non-resonant) then, though no BNF is available, m (t) in (1.15)goestozeroas t goes to zero but not necessarily as a power of t: indeed, in the corresponding (AA) case of sufﬁciently smooth Hamiltonian systems, Bounemoura proves in [6], in any number of degrees of freedom In the (AA) case M ∩{|r| < t}={(θ , r) ∈ T × R, |r| < t} and in the (CC) case M ∩{|r| < t}={(x, y) ∈ R R 2 2 R × R,(1/2)(x + y )< t}. This means one can deﬁne B (f ) for any N ≥ 2. If we denote p /q the convergents of ω,itreads ln q = o(q ). In comparison, the classical Diophantine condition n n n+1 n amounts to ln q = O(ln q ). n+1 n 12 RAPHAËL KRIKORIAN and under a Kolmogorov non-degeneracy condition, that the origin is KAM stable (see [13] for a previous similar result in two degrees of freedom and in the real analytic case) and provides measure estimates for the complement of the set of the invariant tori. Let us add that the twist condition (1.14) in Moser’s Theorem can be considerably weak- ened (see for example [13], [12]). When d = 1, symplecticity (area preservation) can be replaced by the weaker intersection property. When f is real analytic and ω (both in the (CC) and (AA) cases) is Diophantine one can get, by pushing to its limit the “standard” KAM method, a better estimate: for any 0 <β 1and t 1 one has −β 1+τ(ω) (1.16) m (t) exp(−(1/t) ) where we have deﬁned for any irrational ω − ln min |kω − l| ln q l∈Z n+1 (1.17) τ(ω) = lim sup = lim sup ≥ 1; ln k ln q k→∞ n→∞ n in the preceding formula (p /q ) is the sequence of convergents of ω.Notethatif n n n≥0 τ(ω) < ∞,then ω is Diophantine with exponent τ for any τ >τ(ω) (cf. (1.10)). In this case, the inequality (1.16) is known to be true with the exponent 1/(1 + τ ) on the right hand side (see for example [26] and the references therein). If τ(ω)=∞ we say that ω is Liouvillian. 1.3. Optimal and improved measure estimates. — The main results of our paper are that: (A) one can improve the exponent in (1.16)if BNF(f ) converges; (B) in the “general case” the exponent in (1.16) is almost optimal. More precisely Theorem A. — Let f be a real analytic symplectic diffeomorphism f : (R × R,(0, 0)) ý like (1.5)or f : (T × R, T ) ý like (1.6) satisfying the twist condition (1.14) and assume that in both cases ω is Diophantine. Then, if BNF(f ) deﬁnes a converging series one has for any 0 <β 1 and 0 < t 1 (1/τ (ω))−β (1.18) m (t) exp − . On the other hand general real analytic twist symplectic diffeomorphisms like (1.5), (1.6) behave quite differently: Theorem B. — Let ω ∈ R be Diophantine. There exist real analytic twist symplectic diffeo- morphisms f : (R × R,(0, 0)) ý like (1.5)or f : (T × R, T ) ý like (1.6) satisfying the twist I.e. accumulated by a positive measure set of invariant quasi-periodic tori. 24 ∗ As usual, if ω = 1/(a + 1/(a + 1/(··· ))), a ∈ N ,wedeﬁne p /q = 1/(a + 1/(a + 1/(···+ 1/a )). 1 2 i n n 1 2 n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 13 condition (1.14) and a sequence of positive numbers (t ) converging to zero such that for any 0 <β 1, 0 < t 1 k β ( )+β 1+τ(ω) (1.19) m (t ) exp − . f k As we already mentioned in the previous subsection, when ω is very Liouvillian, −1 for example when lim inf (q ln q )> 0, it is not clear, in the (AA) case, how to deﬁne n n+1 BNF(f ). On the other hand, in the (CC) case, BNF(f ) is deﬁned whenever ω is non- resonant and as we will soon see in Theorem A’ below, the result of Theorem A extends to this situation. One might still wonder whether a weaker Diophantine condition ln q = n+1 o(q ) (or something slightly stronger) is enough to ensure the validity of Theorem A in the (AA) case (remember that in this case, BNF(f ) is well deﬁned). It seems possible that adapting Propositions 5.3, 5.5 to this situation (using e.g. [44], [45]) provides estimates that are still good enough to make the proof of Theorem A work. Let us deﬁne for ω ∈ R Q −1 5min(|b (f )|,|b (f )| ) 2 2 (1.20) t (ω) = . q q n+1 n Theorem A’. — Let ω be Liouvillian and f : (R× R,(0, 0)) ý be a real analytic symplectic diffeomorphism of the form (1.5) satisfying the twist condition (1.14). Then, if BNF(f ) deﬁnes a converging series, one has for every k ∈ N large enough such that q ≥ q k+1 1/5 (1.21) m (t (ω)) exp(−q ). f k k+1 Note that if ω is Liouvillian, one has for inﬁnitely many k, q ≥ q . k+1 On the other hand: Theorem B’. — For any ω ∈ R Q, there exist real analytic symplectic diffeomorphisms f : (R × R,(0, 0)) ý of the form (1.5) satisfying the twist condition (1.14) such that for every β> 0 and inﬁnitely many k ∈ N (1.22) m (t (ω)) exp(−q ). f k k+1 Theorem A is a consequence of the following theorem: Theorem C (Small holes).— Let ω be Diophantine and let f : (R× R,(0, 0)) ý of the form (1.5)or f : (T × R, T ) ý of the form (1.6) be a real analytic symplectic diffeomorphisms satisfying the twist condition (1.14). Then, for t > 0, there exists a ﬁnite collection D of pairwise disjoint disks D of the complex plane centered on the real axis such that, for any 0 <β 1, 0 < t 1 one has: q q (1) The number #D of disks in the collection D satisﬁes t t 1−β (1.23)#D (1/t) t 14 RAPHAËL KRIKORIAN and one has −β q q q 1+τ(ω) (1.24) ∀ D ∈ D |D ∩ R| exp(−(1/t) ) (1/τ (ω))−β (1.25) m (t) exp(−(1/t) ) + |D ∩ R|. q q D∈D q q (2) If BNF(f ) converges, then for any t 1 one has for each D ∈ D β t (1/τ (ω))−β (1.26) |D ∩ R| exp(−(1/t) ). Estimate (1.24) explains why in general (without the assumption that BNF(f ) con- verges) one only gets the estimate (1.16). We shall explain in Section 1.5.1 where these disks Dcome from. There is a corresponding theorem in the Liouvillian (CC) case that implies The- orem A’. We shall not state it but we mention that it is a consequence of Theorem 12.6 and Corollary 13.6. Theorems A, A’, C are proved in Section 14 as consequences of Theorems 12.3, 12.6 and Corollaries 13.2, 13.6. Theorems B and B’ are consequences of Theorems E and E’ which are stated in the next Section 1.4. These Theorems are proved in Section 16 which uses results from Section 15. Because 1/τ > 1/(1 + τ), Theorems A–B, A’–B’ clearly imply, in the (AA) and (CC) case, when d = 1, the existence of a diffeomorphism f of the form (1.5)–(1.6) with divergent BNF. We explain in the next Section 1.4, see Theorem D, that this implies Main Theorems 1–1’ in the elliptic ﬁxed point case and the action-angle case for any d ≥ 1 and in a prevalent way. 1.4. Prevalence of divergent BNF’s. 1.4.1. The Dichotomy Theorem. — Let us explain more precisely the dichotomy of R. Pérez-Marco mentioned in Section 1.1 Deﬁnition 1.2. — Asubset A of a real afﬁne space E is (PM)-prevalent if there exists F ∈ A such that for any F ∈ E the set {t ∈ R, tF + (1 − t)F ∈ / A} has 0 Lebesgue measure. 0 0 Pérez-Marco’s dichotomy for Hamiltonians having a non-resonant elliptic ﬁxed point can be reformulated the following way: let E be theafﬁne spaceofrealanalytic See [22] for the concept of prevalence. We can replace zero Lebesgue measure by zero (logarithmic) capacity like in Pérez-Marco’s paper. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 15 Hamiltonians 2 2 3 H(x, y) = 2π ω (x + y )/2 + F(x, y), F(x, y) = O (x, y) j j j=1 which are perturbations of a given non-resonant quadratic part 2 2 (r) = 2πω, r= 2π ω (x + y )/2 ω j j j j=1 and let A be the set of those Hamiltonians which have a divergent BNF. Then, Pérez- Marco’s dichotomy is: either for any H ∈ E ,BNF(H) converges or A is (PM)-prevalent. ω ω We now discuss the extension of Pérez-Marco’s dichotomy to the case of symplectic diffeomorphisms in the (AA) and (CC)-cases. d d Any real analytic symplectic diffeomorphism f : (R × R ,(0, 0)) ý of the form d d (1.5)or f : (T × R , T ) ý of the form (1.6) can be parametrized in the following con- venient form: (1.27) f = ◦ f , 2πω,r F d d 3 d d 2 where, F : (R × R ,(0, 0)) → R,F = O (x, y) or F : (T × R , T ) → R,F = O (r) is some real analytic function and where we denote f : (x, y) → ( x, y) or (θ , r) → (θ, r) the exact-symplectic map (see Section 4.5) deﬁned implicitly by x = x + ∂ F(x, y), y = y + ∂ F(x, y)(CC case) y x (1.28) or θ = θ + ∂ F(θ , r), r = r + ∂ F(θ , r)(AA case). r θ d d d d d For d ≥ 1, ω ∈ R non-resonant, we deﬁne S (R × R ) (resp. S (T × R )) the set of real ω ω d d d d analytic symplectic diffeomorphisms f : (R × R ,(0, 0)) ý (resp. f : (T × R , T ) ý) d d 3 d of the form f = ◦ f with F : (R × R ,(0, 0)) → R,F = O (x, y) (resp. F : (T × 2πω,r F d 2 d d R , T ) → R,F = O (r)) real analytic. We then say that a subset of S (R × R ) (resp. 0 ω d d S (T × R ))is (PM)-prevalent if it is of the form { ◦ f , F ∈ A} for some (PM)- ω 2πω,r F ω d d 3 ω d d 2 prevalent subset A of C (R × R , R) ∩ O (x, y) (resp. C (T × R , R) ∩ O (r)). Here is the version of Pérez-Marco’s Dichotomy Theorem [36] for real analytic symplectic diffeomorphisms of the 2d -disk or the 2d -cylinder. Theorem 1.3 (Dichotomy Theorem).— Let d ≥ 1 and ω ∈ R be a non-resonant frequency d d vector. Then, either for any f ∈ S (R × R ), the formal series BNF(f ) converges (i.e. the series it d d deﬁnes has a positive radius of convergence), or there exists a (PM)-prevalent subset of S (R × R ) such that for any f in this subset BNF(f ) diverges. d d The same dichotomy holds in S (T × R ) provided ω is Diophantine. ω 16 RAPHAËL KRIKORIAN As we mentioned earlier Pérez-Marco’s Dichotomy Theorem was proved in the setting of real analytic Hamiltonians having an elliptic ﬁxed point. Its extension to the dif- feomorphism setting follows essentially Pérez-Marco’s arguments. We refer to Section 6.2 for further details in particular in the Action-Angle case (cf. Lemma 6.3). 1.4.2. Prevalence of the divergence of the BNF: Main Theorems 1, 1’.— As a Corollary of Theorem 1.3 we now obtain, using Theorems A and B,Theorems A’ and B’,the following precise formulation of Main Theorems 1, 1’: d d d Theorem D. — For any d ≥ 1 and any non-resonant ω ∈ R ,the setof f ∈ S (R × R ) d d with a divergent BNF is (PM)-prevalent. If ω is Diophantine the same result holds with S (T × R ) d d in place of S (R × R ). Proof. — We give the proof in the case of real analytic symplectic diffeomorphisms of the 2d -disk. Let ω = (ω ,...,ω ) ∈ R be non-resonant. According to Pérez-Marco’s di- 1 d chotomy (Theorem 1.3) it is enough to provide one example of a real analytic symplectic diffeomorphism of the 2d -disk with diverging BNF and frequency vector ω at the ori- gin to get the conclusion. Since ω is non-resonant, there exists 1 ≤ j ≤ d such that ω is irrational. According to whether ω is Diophantine or Liouvillian we use Theorems A and B or Theorems A’ and B’ to produce a real analytic symplectic diffeomorphism f : (R , 0) ý with frequency ω at the origin and with a divergent BNF. We now deﬁne j j d d f : (R × R ,(0, 0)) ý by f (x ,..., x , y ,..., y ) = ( x ,... x , y ,..., y ), 1 d 1 d 1 d 1 d √ √ 2π −1ω for k = j,( x + −1 y ) = e (x + −1y ) k k k k ( x , y ) = f (x , y ). j j j j j This diffeomorphism is real analytic, symplectic and BNF(f )(r ,..., r ) = BNF(f )(r ) + 2πω r 1 d j j k k k∈{1,...,d}j is diverging since BNF(f ) is. 1.4.3. Prevalence of optimal estimates: Main Theorem 2.— We now present two theo- rems (Theorems E and E’) stating that the measure estimates (1.19)ofTheorem B and (1.22)ofTheorem B’ are prevalent. Together with Theorems A, A’ and the fact that 1/(τ + 1)< 1/τ , this gives a more precise meaning to our Main Theorem 2. We shall treat the (AA) and (CC) cases separately. 2 N 2 ∗ Let X be the set ([−1, 1] ) ={(ζ ,ζ )∈[−1, 1] , k ∈ N } endowed with the 1,k 2,k ⊗N product measure μ = (Leb 2 ) . ∞ [−1,1] ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 17 (AA) Case. Let f = ◦ f 2 be a real analytic symplectic twist map of the annulus of 2πωr O(r ) the form (1.6) and satisfying the twist condition (1.14). For ζ ∈ X and h > 0we deﬁne G ∈ C (T × R) (h > 0ﬁxed) a −|k|h G (θ , r) = r e (ζ cos(kθ) + ζ sin(kθ)) ζ 1,k 2,k k∈N where a is some universal integer (appearing in Proposition G.1 of Appendix G). Theorem E ((AA) case).— For any Diophantine ω and for any 0 <β 1, there exists an inﬁnite set N ⊂ N such that if t = t (ω) is the sequence deﬁned by (1.20), then for μ -almost β k k ∞ ζ ∈ X , the estimate (1.19)ofTheorem B with f replaced by f is satisﬁed for inﬁnitely many k ∈ N . ζ β In particular, using Theorem A,the BNFof f := f ◦ f is divergent for μ -almost ζ ∈ X . ζ G ∞ (CC) Case. We formulate the corresponding result in the (CC) Case in a less general setting than in the (AA) Case. We assume that 2 2 a f = 2 2 + O((x + y ) ) ((1/2)(x +y )) where −1 2 3 (2π) (r) = ωr + b r + O(r ), b = 0. 2 2 We shall denote sign(b )=±1if ±b > 0. 2 2 For ζ ∈ X ,let G be the real analytic function a q q 2 2 3 k k x + y ζ x + iy x − iy 1,k ∗ −iπ/4 iπ/4 G (x, y) = × √ e + √ e 2 2 2 2 k=1 q q k k ζ x + iy x − iy 2,k −iπ/4 iπ/4 + × √ e − √ e 2i 2 2 and f = f ◦ . ζ G Theorem E’ ((CC) Case).— For any non-resonant (resp. Diophantine) ω and any 0 <β 1, there exist a non-empty set s(ω) ∈{−1, 1} (resp. s (ω) ∈{−1, 1}) and an inﬁnite set N ⊂ N (resp. N ⊂ N) such that the following holds. If sign(b ) ∈ s(ω) (resp. sign(b ) ∈ s (ω)), then, for 2 2 β μ -almost ζ ∈ X , the estimate (1.22)ofTheorem B’ (resp. (1.19)ofTheorem B) with f replaced by f is satisﬁed for inﬁnitely many k ∈ N (resp. k ∈ N ). In particular, using Theorems A, A’, for any non-resonant ω,the BNFof f := f ◦ is divergent for μ -almost ζ ∈ X . ζ G ∞ We refer to Section 16 for the proof of Theorems E and E’. In the non-resonant case, the sets s(ω) and N ⊂ N do not depend on β . 18 RAPHAËL KRIKORIAN 1.5. Some words on the proofs. — The starting point of the proofs of Theorems A, A’,and C is a KAM scheme that we implement on a holomorphic extension of the real analytic diffeomorphism f . This allows to work with holomorphic functions deﬁned on complex domains “with holes” (i.e. disks which are removed). If these domains are “nice” we can use some quantitative form of the analytic continuation principle to propagate informations in the neighborhood of the origin, like the convergence of the BNF, to the neighborhoods of each hole. We illustrate this with the proof of Theorem C. 1.5.1. Sketch of the proof of Theorem C.— We describe it in the (AA) case. Let f : (T × R, T ) ý, T = T×{0}, be a real analytic symplectic diffeomorphism of the form 0 0 (1.27)with b (f ) = 0and ω Diophantine. After performing some steps of the Birkhoff Normal Form procedure mentioned in the introduction, we can assume that −1 2 m (1.29) f = ◦ f ,(2π) (r) = ωr + b r +··· , F = O(r ) F 2 where m is large enough and where f is the exact symplectic map (cf. (1.28)) associated to some real symmetric holomorphic function F : T × D(0, ρ) → C (h,ρ> 0); the notations T , D(0, ρ) are for T := ((R+ i]− h, h[)/(2π Z)), D(0, ρ)={r ∈ C, |r| < ρ}. h h Adapted KAM Normal Form. — Theorem C can be seen as an improved version of the classic KAM Theorem on the positive Lebesgue measure of the set of points lying on invariant curves (cf. Moser’s Theorem of Section 1.2). There are several ways to prove this standard KAM Theorem. A direct approach (which goes back to Arnold in his proof of Kolmogorov’s theorem) is to ﬁnd a sequence of (real symmetric) holomorphic symplectic diffeomorphisms g close to the identity, deﬁned on smaller and smaller complex domains −1 T × U (h ≥ h ≥ h/2, U ⊂ U ⊂ D(0, ρ))and such that g ◦ f ◦ g gets closer and h i i−1 i i i−1 i i i closer to some integrable models : −1 (1.30) [T × U ] g ◦ f ◦ g = ◦ f , F 1 h i i F i i i i i (in the preceding formula, the set written on the left is a domain where the conjugation relation holds); see Figure 1. One then proves that g and converge (in some sense) i i on T × (U ∩ R) (U := U ) to some limits g , and that U ∩ R (in general a ∞ ∞ i ∞ ∞ ∞ Cantor set) has positive Lebesgue measure. The searched for set of f -invariant curves is then g ({r = c}) and one has for some ﬁxed constant a > 0and any ρ< ρ c∈U ∞∩R (1.31) m (ρ) F . We refer to Theorem 12.1 for more details. The domains U can be chosen to be holed domains i.e. disks D(0,ρ )(ρ ≈ ρ ) from which a ﬁnite number of small complex disks i i This means that it takes real values when θ and r are real. This means that depends only on the r variable. i ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 19 D(0, ρ) R-axis KAM FIG. 1. — The holed domains U where the KAM-Normal Form U is deﬁned (the holes, caused by resonances, are the grey disks) centered on the real axis (the “holes” of U ) have been removed. Removing these small disks is due to the necessity of avoiding resonances when one inductively construct g , , F i i i from g , , F . More precisely, U is essentially obtained from U by removing i−1 i−1 i−1 i i−1 −1 “resonant disks” i.e. disks where the “frequency map” (2π) ∂ is close to a ratio- i−1 nal number of the form l/k, (l, k) ∈ Z × N ,max(|l|, k) N (N is an exponentially i−1 i increasing (in i) sequence which is deﬁned at the beginning of the inductive procedure). The sizes of the holes of U created by removing a ﬁnite number of disks from U decay i i−1 very fast with i. We shall call a conjugation relation like (1.30) a(n) (approximate) KAM Normal Form for f . Its construction is presented in Section 7. A useful observation (cf. Section 10) is that, depending on ρ< ρ , one can choose indices i (ρ) < i (ρ) such that all the holes D of the domain U that intersect D(0,ρ), − + i (ρ) are disjoint and are created at some step i − 1 = i ∈[i (ρ), i (ρ)] (hence D ⊂ U ); D − + i moreover, i (ρ) is large enough to ensure that the size of D is small. Writing (1.30)with i = i (ρ) we get (note the change of notations) KAM −1 KAM KAM KAM (1.32) [T × U ] g ◦ f ◦ g = ◦ f , F 1. h/2 i (ρ) i (ρ) i (ρ) + i (ρ) + + + i (ρ) i (ρ) + + This is what we call our adapted KAM Normal Form (adapted to D(0,ρ)); see Section 10. With the choice we make for i (ρ) we have KAM (1/τ )− (1.33) F exp(−(1/ρ) ), i (ρ) KAM (1/τ )−β where the last formula means: “for any β> 0, F exp(−(1/ρ) )”. i (ρ) Hamilton-Jacobi Normal Forms. — Cf. Section 8.Ahole D ⊂ U of the domain U that i i (ρ) D + −1 KAM is created at step i corresponds as we have mentioned to a resonance (2π) ∂ ≈ l/k, (l, k) ∈ Z × N ,max(|l|, k) N that appears when one constructs the KAM Nor- mal Form (1.30)fromstep i to step i + 1. In this resonant situation we are able to D D 20 RAPHAËL KRIKORIAN HJ −1 KAM FIG. 2. — Hamilton-Jacobi Normal Form close to a resonance (2π) ∂ (c) = l/k. The holomorphic function D i HJ is deﬁned on the annulus D D associate to D a Hamilton-Jacobi Normal Form, cf. Section 8,Proposition 8.1: there exists an q q q annulus D D(D, D are disks), D ⊂ U , D ⊃ D, D ⊃ D(D is small but much bigger than D) on which one has HJ HJ −1 HJ HJ (1.34) [T × D D] (g ) ◦ ◦ f ◦ (g ) = ◦ f , h/9 F D D i i F D D D D HJ KAM (1.35) F F . D i (ρ) See Figure 2. This HJ Normal Form also satisﬁes the important Extension Property which in q q some situation allows to bound above the size of D (note that in general the sizes of Dand HJ D are comparable). It states that if the holomorphic function , which is deﬁned on the annulus D D, coincides to some very good order of approximation with a bounded holomorphic function deﬁned on the disk D, then D can be chosen to be small (see the quantitative statement of Proposition 8.1). Proof of the ﬁrst part (1.25)ofTheorem C.— Applying the aforementioned standard KAM estimate (1.31) on the holed domain U to ◦ f KAM (cf. (1.32)) and on each annulus i (ρ) + i (ρ) F + i (ρ) D Dto HJ ◦ f HJ,(cf. (1.34)) together with the estimate (1.35) we get that outside a D D set of measure |D ∩ R| the invariant curves of f cover a set the complement of D∈D KAM a which in D(0,ρ) has a measure F for some a > 0; hence the inequality (1.25) i (ρ) by (1.33). For more details see the proof of Theorem 12.3. Birkhoff Normal Forms. — Cf. Section 6. To prove the second part of Theorem C,(1.26)we need to introduce one further approximate Normal Form, namely the approximate Birkhoff b b KAM τ τ Normal Form (cf. Section 6)valid on T × D(0,ρ ) (b = τ + 1), D(0,ρ ) ⊂ U h/2 τ i (ρ) b BNF −1 BNF (1.36) [T × D(0,ρ )] (g ) ◦ f ◦ (g ) = BNF ◦ f BNF , h/2 F ρ ρ ρ ρ BNF KAM (1.37) F F . ρ i (ρ) See Figure 3. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 21 KAM D(0,ρ) i (ρ) D(0,ρ ) R-axis BNF b FIG. 3. — The approximate Birkhoff Normal Form is deﬁned on D(0,ρ ). It coexists with the KAM Normal Form KAM deﬁned on the holed domain U i (ρ) i (ρ) + KAM D(0,ρ) i (ρ) D(0,ρ ) R-axis D, D BNF b FIG. 4. — The three Normal Forms. The holomorphic function is deﬁned on D(0,ρ ); associated to each hole D HJ the holomorphic function is deﬁned on the annulus D D. These Normal Form coincide with the KAM Normal KAM KAM Form deﬁned on the holed domain U i (ρ) i (ρ) + + Proof of the second part (1.26)ofTheorem C.— Having the three Normal Forms (1.32), (1.36), (1.34) in hand (see Figure 4) the proof of the second part of Theorem C relies on the following three principles. 22 RAPHAËL KRIKORIAN HJ KAM BNF – Comparison Principle cf. Section 9: since F ,F F are equally very small, all the i (ρ) D ρ previous Normal Forms almost coincide on the intersections of their respective domains of deﬁnition (this is done in Proposition 9.1); more precisely their frequency maps almost coincide HJ BNF KAM (1.38) ρ i (ρ) b KAM KAM τ q D(0,ρ )∩U U ∩(DD) i (ρ) + i (ρ) where the symbol a b (a, b are functions and V is an open set) means here: for all z ∈ V, (1/τ )− |a(z) − b(z)| exp(−(1/ρ) ), ρ and τ being ﬁxed. Moreover, if the formal BNF converges and equals a holomorphic function deﬁned on, say, D(0, 1), one has also (cf. Corollary 6.7) BNF D(0,1)∩D(0,ρ ) b KAM and in particular from (1.38)(we have D(0,ρ ) ⊂ U ) i (ρ) KAM (1.39) . i (ρ) D(0,1)∩D(0,ρ ) KAM b – No-Screening Principle, cf. Section 3: Since and almost coincide on D(0,ρ ) and i (ρ) KAM are holomorphic on the bigger domain D(0, 1)∩ U , one can be tempted to infer that i (ρ) they also almost coincide on this latter domain. A difﬁculty could appear here: an exces- KAM sive number of holes of D(0, 1) ∩ U (in comparison to their sizes) could cause some i (ρ) “screening effect” (like in Electrostatics) that prevents the propagation of the information KAM given by (1.39) to “most of ” the domain D(0, 1)∩ U ; see Section 3.2 for more details. i (ρ) This is the reason why, instead of working on the whole domain D(0, 1) we work on the smaller one D(0,ρ). In this situation, the choice we make for i (ρ) (cf. (10.301)) is such KAM that the number of holes of D(0, 1) ∩ U is not too big in comparison to their sizes; i (ρ) this is studied in Section 10,Proposition 10.4. This allows us to apply Proposition 3.1 and to extend the domain of validity of the approximate equality (1.39) to (a good part KAM of) D(0, 1) ∩ U : i (ρ) KAM (1.40) . i (ρ) KAM D(0,1)∩U i (ρ) – Residue or Extension Principle cf. Section 8.8: From (1.38), (1.40) one has HJ (1.41) KAM U ∩(DD) i (ρ) HJ or,inother words, , which is deﬁned on the annulus D D, coincides with a very good approximation with a holomorphic function deﬁned on the whole disk D. The aforementioned Extension Principle of Proposition 8.1, which essentially amounts to the computation of ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 23 a residue (done in Paragraph 8.8.1), then tells us that the radius of D is much smaller (1/τ )− than what we expected it to be: ﬁnally, |D ∩ R| exp(−(1/ρ) ). This is (1.26). Formoredetails we refertoProposition 10.7 and Corollary 13.2. 1.5.2. On the proof of Theorem C in the elliptic ﬁxed point case. — The proof in the non- resonant elliptic ﬁxed point case, f : (R , 0) ý, follows the same strategy especially if the frequency ω is Diophantine. A technical point is that to be able to implement the KAM BNF No-Screening Principle of Section 3 we need to work with domains U ⊃ U where i (ρ) ρ BNF U is a disk around 0 (the estimate on the analytic capacity of this disk is then favorable). This is the reason why we cannot in this situation use Action-Angle variables since this would force us to work on angular sector domains and not disks. Instead, we deﬁne our approximate BNF and KAM Normal Forms directly in Cartesian Coordinates. The formalism turns out to be the same as in the Action-Angle case (see Section 5), so we treat these two cases simultaneously. The case where ω is Liouvillian is done in a similar (and even simpler) way. 1.5.3. On the proofs of Theorems B, B’, E and E’.— The proofs are more classical and based on the fact that, in the general case, resonances are associated to the existence of hyperbolic periodic points in the neighborhood of which no (“horizontal”) invariant circle can exist. To see this in a special situation (we describe it in the (AA)-case) let f = ◦ f , −1 2 ∗ where R r → (2π) (r) = (p/q)r + br /2 + ··· ∈ R with p ∈ Z, q ∈ N mutually prime and, say, b > 0 (for example b = 1); we also assume that T× R (θ , r) → F(θ , r) = 3 −1 O(r ) ∈ R and is (2π/q)-periodic in θ . The origin is thus resonant since (2π) ∂(0) = p/q is rational. One can approximate ◦ f by per 1 f := (θ , r) → 2π(p/q, 0) + (θ , r) 1 2 where is the time-1 map of the Hamiltonian H(θ , r) = br /2 + F(θ , r). Observe that because we have assumed that F(θ , r) is 2π/q-periodic in θ , the same is true for , hence the maps and (θ , r) → (θ , r) + 2π(p/q, 0) commute; thus, understanding the dynamics of f essentially amounts to understanding that of . This latter dynamics is easy to analyze since it is the time-1 map of a Hamiltonian vector ﬁeld in dimension 2, namely a pendulum on the cylinder T× R. Indeed, if F is “typical”, a change of coordinates leadsustothe case where ∂ F(0, 0) = 0and ∂ F(0, 0)< 0hence H(θ , r) = cst+ br /2− (1/2)|∂ F(0, 0)|θ + h.o.t. Under this form it is clear that (0, 0) is a hyperbolic ﬁxed point for and since H is 2π/q-periodic, the same is true for the points (2π k/q, 0), k = 0,..., q − 1. Since these points are permuted by (θ , r) → (θ , r) + 2π(p/q, 0), this shows that (0, 0) is a hyperbolic q-periodic point for f (this means a hyperbolic ﬁxed q 2 per point for f ). If |∂ F(0, 0)| is not too small compared to the approximation f − f ,the To say it shortly, in Poisson-Jensen’s formula on subharmonic functions (see Section 3.1), the “weight” of a small disk D(0,ρ) ⊂ D(0, 1) is 1/| ln ρ| while the “weight” of D(0,ρ)∩ ⊂ , being an angular sector at 0 is only ρ , a > 0. 24 RAPHAËL KRIKORIAN point (0,0) will also be a q-periodic hyperbolic point for f . However, a horizontal invariant circle cannot cross the stable or invariant manifolds of this periodic point; this establishes the existence of a zone in which horizontal invariant circles cannot pass. To quantify the size of this zone one just has to estimate the strength of the hyperbolicity of the periodic point and the size of the corresponding local stable and unstable manifolds. The more general case where we do not assume a priori that F(θ , r) is 2π/q-periodic nor F(θ , r) = O(r ) can essentially be reduced to the preceding example, provided F is small with respect to 1/q. This requires the use of a resonant normal form described in Appendix G. For “generic” symplectic diffeomorphisms of the form (1.6) satisfying a twist condition (1.14) one can establish the existence of hyperbolic zones associated to any best rational approximation p /q of ω; these zones accumulate the origin. We refer n n to Sections 15 and 16 for more details. 1.6. Organization of the paper. — Section 2 is essentially dedicated to ﬁxing some notations and introducing the notion of domains with holes that plays a central role in the KAM approach (à la Arnold). We discuss Cauchy’s estimates and Whitney’s extension Theorem in this framework. The not so standard notations used in the text are summa- rized in Section 2.6. In Section 3 we give a brief account of what is the screening effect and we provide a no-screening criterion which will be useful for our purpose. It is based on Poisson-Jensen’s formula on subharmonic functions applied in a domain with not too many holes (w.r.t. their sizes). In Section 4 our main purpose is to check that estimates on compositions of gener- ating functions hold in the case of domains with holes. We treat in a uniﬁed way the CC and AA cases. We also discuss invariant curves. In Section 5 we study the (co)homological equations and state a proposition on the basic KAM step (Proposition 5.5). Birkhoff Normal Forms (approximate and formal) are presented in Section 6 and Appendix E. We explain in Section 6.2 how Pérez-Marco’s dichotomy extends to the diffeomorphism case. Section 7 is dedicated to the KAM scheme which is central in our paper; we pay particular attention to the location of the holes of the KAM-domains. In Section 8 we present the Hamilton-Jacobi Normal Form associated to each resonance appearing during the KAM scheme. Their construction is based on a Res- onant Normal Form and an argument of approximation by vector ﬁelds the proofs of which are left in the Appendix, Sections G and H. The most important property of these Hamilton-Jacobi Normal Forms is the Extension Property that states that if the correspond- ing frequency map deﬁned on a annulus is very close to a holomorphic function deﬁned on a bigger disk containing the annulus, the domain of validity of this Normal Form is essentially this disk. The Matching or Comparison Principle is presented in Section 9. It quantiﬁes the fact that (exact) symplectic maps have essentially one frequency map. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 25 Residue/Extension No screening Principle (Section 3) (Section 8) HJ-NF≈ HJ-NF on Dsmall on D D D D (Section 8) (Sections 8, 13) (Section 8) Resonant NF (Appendix G) Resonances, Holes D ∈ D (Section 8) Positive KAM measure procedure estimates (Section 7) (Section 12) Matching KAM NF on Principle U = D(0,ρ) D D∈D (Section 9) (Sections 7, 10)) Formal BNF Improved Approx. BNF converges measure bτ on D(0,ρ ) ≈ = on D(0, 1). estimates (Section 6) (Assumption) (Conclusion) FIG. 5. — Plan of the proof of the improved measure estimate (1.26)ofTheorem C We construct in Section 10 and 11 our coexisting adapted KAM, BNF and HJ Normal Forms in the respective cases ω Diophantine or Liouvillian, the latter being easier to treat. In Section 12 we ﬁrst state a generalization of the classical KAM estimate on the measure of the set of invariant curves that hold on domains with holes (Theorem 12.1) and we apply it to our adapted KAM and HJ Normal forms to get measure estimates on the set of invariant curves lying in the union of the domains of deﬁnitions of these Normal Forms. This provides Theorems 12.3 and 12.6 which play an important role in the proofs of Theorems C, A and A’. 26 RAPHAËL KRIKORIAN In Section 13 we use the Extension Principle of Section 8 to show that if the BNF converges the measure estimates provided by Theorems 12.3 and 12.6 improve consid- erably. In Section 14 we conclude the proofs of Theorems C, A and A’. The mechanism for the creation of zones of the phase space that do not intersect the set of invariant circles is presented in Section 15 (Proposition 15.1). This allows us to construct (prevalent) examples that satisfy Theorems B, B’, E and E’ in Section 16. Finally, an Appendix completes the text by giving more details on the proofs of some statements or by presenting more or less classical methods that had to be adapted to our more speciﬁc situation. 2. Notations, preliminaries Let T be the 1-dimensional torus T := R/(2π Z)={x + 2π Z, x ∈ R} and for 0 ≤ h≤∞ T = T∪{x + iy + (2π Z), x, y ∈ R, |y| < h} (i =−1) the complex cylinder of width 2h.If θ = (x + iy )+ (2π Z), θ = x + iy + (2π Z) ∈ T 1 1 1 2 2 2 ∞ we set |θ − θ | := min |(x − x − 2π l) + i(y − y )|. 1 2 T l∈Z 1 2 1 2 If ρ> 0 wedenoteby D(z,ρ) ⊂ C the open disk of center z and radius ρ and by D(z,ρ) its closure; sometimes for short we shall write D for D(0,ρ) (and by D its ρ ρ closure). If z = x + iy ∈ C,(i = −1) x, y ∈ R, (resp. θ = x + iy + (2π Z) ∈ T ), we denote by z = x − iy (resp. θ = x − iy + (2π Z)) its complex conjugate. 2 2 We deﬁne the involutions σ ,σ : C → C and σ : T × C → T × C by 1 2 3 ∞ ∞ (2.42) σ (x, y) = (x, y), σ (z,w) = (iw, iz), σ (θ , r) = (θ, r). 1 2 3 For w = (w ,w ), w = (w ,w ) ∈ C × C (resp. ∈ T × C) we deﬁne the distance 1 2 ∞ 1 2 d(w, w ) = max(|w − w |,|w − w |) (resp. d(w, w ) = max(|w − w | ,|w − w |)). 1 2 1 T 2 1 2 1 2 If W is an open subset of C × C or of T × C and if F : W → C we set F = sup|F| (with the convention that F = 0 if W is empty). If a function W (w ,w ) → W 1 2 F(w ,w ) is differentiable enough (for the standard real differentiable structure on W) 1 2 32 k 1 k 2 1 2 we can as usual deﬁne its partial derivatives ∂ ∂ ∂ ∂ F(k , k , l , l ∈ N) and its 1 2 1 2 w w w w 1 1 2 2 With this notation D(z, 0)=∅. 32 2 Here we use the standard notation: if w = t + is, (t, s) ∈ R , ∂ = (1/2)(∂ − i∂ ) and ∂ = ∂ = (1/2)(∂ + i∂ ). w t s w w t s ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 27 l 2 j k 1 k 1 2 (total) j -th derivative D F = (∂ ∂ ∂ ∂ F) (j ∈ N). We then deﬁne k +k +l +l =j w w w w 1 2 1 2 1 1 2 2 j k l k j 1 1 2 2 D F = max ∂ ∂ ∂ ∂ F , F n = maxD F . W W C (W) W w w w w 1 1 2 2 0≤j≤n (k ,l ,k ,l )∈N 1 1 2 2 k +l +k +l =j 1 1 2 2 We denote by C (W) the set of functions F : W → C such that F n < ∞ and by C (W) O(W) the set of holomorphic functions F : W → C (all the preceding partial derivatives of the form ∂ = ∂ then vanish). w w We say that an open set W of M := C or of M := T × C is σ -symmetric (i = ∞ i 1, 2, 3) if it is invariant by σ (σ (W) = W); if W is σ -symmetric we say that a function i i i F : W → C is σ -symmetric if F ◦ σ = F (the complex conjugate of F) and we denote i i n n by C (W), resp. O (W), the set of C resp. holomorphic functions F : W → C that are σ i σ -symmetric. When no confusion is possible on the nature of the relevant σ involved, we i i shall often say σ -symmetric or even real symmetric instead of σ -symmetric. If W is σ - symmetric we use the notation W ={w ∈ W,σ(w) = w};if W = ∅ then F ∈ O (W) R R σ deﬁnes by restriction a map (still denoted by F) F : W → R. Note that a function F : (R , 0) → R which is real analytic is in O (D(0,ρ) × D(0,ρ)) for some ρ> 0. 2 n Let W be a open set of M := C or T × C.Wedenoteby Diff (W), resp. O n Diff (W), the set of C , resp. holomorphic, diffeomorphism f : W → f (W) ⊂ Mdeﬁned on an open neighborhood W of W containing the closure of W. Note that there exists a constant C depending only on M such that for any C - diffeomorphisms f , f : M → M satisfying f − id 1 ≤ 1 one has 1 2 1 C (2.43) (f ◦ f ) − id 1 ≤ C(f − id 1 +f − id 1 ). 2 1 C 1 C 2 C If now W is a σ -symmetric open set of (M,σ ) we denote by Diff (W) resp. O n O Diff (W) the set of f ∈ Diff (W), resp. f ∈ Diff (W),suchthat f ◦ σ = σ ◦ f .Itthen deﬁnes by restriction a C , resp. real analytic, diffeomorphism (that we still denote f ) f : W → f (W ) ⊂ M . R R R When f , g are two σ -symmetric holomorphic diffeomorphisms we write (2.44) [W] f = g (W possibly empty) to say that f , g ∈ Diff (W) coincideonanopenneighborhood of W containing the closure of W. 2.1. Domains W .— Let h ≥ 0 and U an open connected set of C; we shall deﬁne h,U AA AA CC CC∗ domains W of M = M = T × C (AA stands for “Action-Angle”) and W ,W h,U h,U h CC CC∗ 2 of M = M = M = C (CC for “Cartesian Coordinates”) the following way: CC∗ – Cartesian Coordinates (CC∗):if ρ := sup{|r|, r ∈ U}, the set W ⊂ C × C is h,U 2 2 x + y 1/2 CC∗ 2 h (2.45)W ={(x, y) ∈ C , |x ± iy|≤ 2e ρ , ∈ U}; h,U U 2 28 RAPHAËL KRIKORIAN R/(2π Z) AA FIG.6.—Thedomain W = T × U h,U 1/2 max(|z|,|w|) ≤ e ρ −izw ∈ U CC CC∗ 1 FIG. 7. — Schematic representation of the domain W (and W if one makes the change of coordinates z = (x + iy), h,U h,U w = (x − iy)) CC – Cartesian Coordinates (CC):if ρ := sup{|r|, r ∈ U}, the set W ⊂ C × C is h,U 1/2 1/2 CC h h (2.46)W ={(z,w) ∈ D(0, e ρ ) × D(0, e ρ ), −izw ∈ U}; h,U U U AA – Action Angle coordinates (AA): the set W of T × C is h,U AA (2.47)W = T × U. h,U 2 2 In all these three cases we denote by r the observable (x, y) → (1/2)(x + y ), (z,w) → −izw, (θ , r) → r. In Section 4.1 we shall see how one goes from (CC) (or (CC*)) to (AA) coordinates. 2.2. Cauchy estimates. — If δ> 0 wedenoteby U (W) ={w ∈ W, B(w, δ) ⊂ W} (here B(w, δ) is the ball {z ∈ M, d(z,w) < δ}). Assume that F ∈ O(W). By differentiating (k + k )- times Cauchy complex integration formula 1 2 1 F(ζ ,ζ ) 1 2 F(w ,w ) = dζ dζ 1 2 1 2 (2π i) (w − ζ )(w − ζ ) 1 1 2 2 |w −ζ |=δ |w −ζ |=δ 1 1 2 2 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 29 one sees that if U (W) is not empty k k −(k +k ) 1 2 1 2 (2.48) ∂ ∂ F ≤ C δ F . U (W) k ,k W w w δ 1 2 1 2 2.3. Holed domains. 2.3.1. Holed domain of C.— A holed domain of C is an open set of C of the form (2.49)U = D(c,ρ) D(c ,ρ ), i i i∈I for some c ∈ C, ρ> 0, c ∈ C, ρ > 0 and where I is a ﬁnite set which is either empty i i or such that for any i ∈ I, D(c ,ρ ) ∩ D(c,ρ) = ∅. Note that the disks D(c ,ρ ) are not i i i i supposed to be included in D(0,ρ). It is not difﬁcult to see that there exists a unique minimal J ⊂ I (for the inclusion) such that D(c ,ρ ) = D(c ,ρ ) and that the U i i i i i∈J i∈I representation (2.49) with I replaced by J is then unique: (2.50)U = D(c,ρ) D(c ,ρ ). i i i∈J We then denote by (2.51) D(U)={D(c ,ρ ), i ∈ J }. i i U We shall call D(c,ρ) the external disk of U. We then set ρ := rad U := ρ, rad(U) = min ρ U i∈J i 2 1/2 a(U) = ( ρ ) i∈J (2.52) card(U) = #J ⎪ U d(U) = rad(U) if J =∅, d(U) = min(rad(U), rad(U)) if J = ∅. U U If J is empty or if all the disks D(c ,ρ ), i ∈ J , are pairwise disjoint and included in U i i U D(c,ρ) we say that the holed domain U has disjoint holes and we call D(c ,ρ ) the holes i i of U (the bounded connected components of C U). We denote by D(U) the set of all these disks. Note: We shall only consider in this paper holed domains (2.49)where the c are on the real axis. 2.3.2. Holed domains of C × C or T × C.— These are by deﬁnition sets of the form W where h > 0 and U is a holed domain; see (2.46)or(2.47). We then deﬁne h,U d(W ) = min(h, d(U)). h,U 30 RAPHAËL KRIKORIAN −δ 2.3.3. Deﬂation of a holed domain. — If δ ∈ R we use the notation e D(c,ρ) for −δ −δ e D(c,ρ) = D(c, e ρ). −δ If U ⊂ C is a holed domain of the form (2.50) and if δ> 0 wedenoteby e U ⊂ Uthe (possibly empty) open set −δ −δ δ e U = D(c, e ρ) D(c , e ρ ). i i i∈J Similarly if 0 <δ < h −δ e W = W −δ . h,U h−δ/2,e U We make the following simple observations (the ﬁrst two items are proved by area considerations): Lemma 2.1. — For 1 >δ > 0 one has: (1) For any z ∈ D(c,ρ), dist(z, U) ≤ 2 a(U). 2 4δ 2 −δ (2) If ρ > 2e ρ then e U is not empty. i∈J i −δ −δ (3) If e U is not empty, then for any z ∈ e U one has D(z,(1/2)δ d(U)) ⊂ U. 2.3.4. Reformulation of Cauchy’s Inequalities. — Using item 3 of Lemma 2.1 we can in particular reformulate inequalities (2.48) when W is of the form W and F ∈ O(W ): h,U h,U m −m −m (2.53) D F −δ ≤ C δ d(W ) F . e W m h,U W h,U h,U One can sometimes obtain better estimates. – In the (AA)-case, if 0 <δ < h, one has k −k (2.54) ∂ F −δ δ F e W W θ h,U h,U −δ – In the (CC)-case, if U = D(0,ρ) and δ< 1/2 one has e W ⊂ U (W ) h,D(0,ρ) δ h,D(0,ρ) 1/2 −h with δ = ρ e δ/4and thus h −1 −1/2 −δ (2.55) ∇ F e δ ρ F . e W W h,U h,U 2.4. Whitney type extensions on domains with holes. — The discussion that follows will be useful in the construction of the KAM Normal Form of Section 7. Let U be a real symmetric holed domain (2.56)U = D(0,ρ) D(c ,ρ ), c ∈ R, i i i i∈J U ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 31 k 33 h > 0, W one of the domains deﬁned in Section 2.1 and F : W → C be a C h,U h,U σ -symmetric function i.e. F ◦ σ = F (the complex conjugate of F). We say that a C , 34 Wh 35 σ -symmetric function F : W → C is a Whitney extension for (F, W ) if h,C h,U Wh ∀ m ∈ W , F (m) = F(m). h,U j j Wh Note that since U is open this implies that for all 0 ≤ j ≤ k,D Fand D F coincide on W . h,U We shall construct such Whitney’s extensions in two situations. Lemma 2.2. — Let F ∈ O (W ). For any δ ∈]0, 1[, there exists a C , σ -symmetric func- σ h,U Wh tion F : W → C such that h,C −δ Wh (2.57) ∀ m ∈ e W , F (m) = F(m) h,U j Wh k −2k j (2.58) sup D F ≤ C(1 + #J ) (δ d(U)) maxD F . W U W h,C −δ/10 h,e U 0≤j≤k 0≤j≤k Proof. — See Section B.1 of the Appendix. Notation 2.3. — We denote by O (W ) the set of C , σ -symmetric maps F : W → C σ h,U h,C such that the restriction of F on W is holomorphic. h,U Deﬁnition 2.4. — Let A ≥ 1, B ≥ 1, U ⊂ C a σ -symmetric holed domain. We say that a σ -symmetric C function : U → C satisﬁes an (A, B)-twist condition on U if 1 1 −1 2 3 (2.59) ∀ r ∈ U ∩ R, A ≤ ∂ (r) ≤ A, and D ≤ B. 2π 2π If U is a disk D(0,ρ ) one can construct for some 0 < ρ< ρ aC , σ -symmetric 0 0 Whitney extension for on D(0, ρ) that satisﬁes an (A, B)-twist condition on D(0, ρ). Lemma 2.5. — Let ∈ O (D(0,ρ )) (ρ ≤ 2) σ 0 0 −1 2 3 (2π) (z) = ω z + b z + O(z ), ≤ 1, with b > 0. 0 2 D(0,ρ ) 2 3 Wh There exists 0 < ρ< ρ , B ≥ 0 and a C , real symmetric extension ∈ O (C) of (, D(0, ρ)) 0 σ −1 that satisﬁes an (A, B)-twist condition on C with A = 3max(b , b ). Proof. — See Appendix B.2. Notation 2.6. — We denote by TC(A, B) the set of C , real symmetric maps : C → C satisfying an (A, B)-twist condition (2.59) with U = C. Differentiability here is related to the real differentiable structure of W . h,C The exponent Wh stands for “Whitney”. See [55], [50]. 32 RAPHAËL KRIKORIAN Let U ⊂ C be a σ -symmetric connected holed domain as in (2.56). Proposition 2.7. — If ∈ O (U) ∩TC(A, B) with (2.60)8 × max(ρ , a(U)) × A × B < 1 2 −1 then the following holds. For any ν ∈]0,(6A B) [ and any β ∈ R, either for any z ∈ U −1 (2.61) |ω(z) − β|≥ν(ω = (2π) ∂) or there exists a unique c ∈] − ρ − 2Aν, ρ + 2Aν[ such that ω(c ) = β and for any z ∈ U β β D(c , 3Aν) one has |ω(z) − β| >ν. Proof. — See Appendix B.3. 2.5. Notation O .— Let h > 0, U be a holed domain, functions F ,..., F ∈ p 1 n O(W ) and l ∈ N . We deﬁne the relation h,U G = O (F ,..., F ) l 1 n as follows: there exist a ∈ N ,C > 0and Q(X ,..., X ) a homogeneous polynomial 1 n (independent of U) of degree l in the variables (X ,..., X ) such that for any 0 <δ < h/2 1 n satisfying −a −a (2.62)Cd(W ) δ maxF ≤ 1 h,U i W h,U 1≤i≤n −δ one has G ∈ O(e W ) and h,U −a −a (2.63) G −δ ≤ d(W ) δ Q(F ,...,F ). e W h,U 1 W n W h,U h,U h,U We shall use the notation O (F ,..., F ) if the polynomial Q is null when X = 0 l 1 n 1 i.e. Q(0, X ,..., X ) = 0; for example if l = n = 2, Q(X , X ) = X X + X . 2 n 1 2 1 2 When we want to keep track of the exponent a appearing in (2.62), (2.63)weshall (a) use the symbol O . When δ satisﬁes (2.62) we write a,C (2.64) δ = d (F ,..., F ; W ) 1 n h,U and we use the short hand notation (2.65) δ = d(F ,..., F ; W ) 1 n h,U to say that (2.64) holds for some positive constants a, C large enough and independent of F ,..., F ,d(W ). 1 n h,U ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 33 Remark 2.1. —Note that if U = D(0,ρ ) is a disk containing 0 and F ∈ O(W ), h,U ∗= CC, AA, one has p p/2 F(z,w) = O (z,w) ⇐⇒ ∀ 0 ≤ ρ ≤ ρ , F ρ 0 W h,D(0,ρ) p p F(θ , r) = O (r) ⇐⇒ ∀ 0 ≤ ρ ≤ ρ , F ρ 0 W h,D(0,ρ) CC AA p hence if F ,..., F ∈ O(W ) (resp. ∈ O(W ))satisfy F (z,w) = O (z,w) (resp. F = 1 n i i h,U h,U O (r)), 1 ≤ i ≤ n, one has mp−2a mp−a O (F ,..., F ) = O (z,w) (resp. O (r)). m 1 n 2.6. Summary of the various notations used in the text. – a b, a b, a b, a b, a betc. See Notation 1.1. β β – a exp(b−) means: for all β> 0, one has a exp(b − β). – A(z; λ ,λ )={w ∈ C,λ < |w − z| <λ }. 1 2 1 2 – T ={x + iy + (2π Z), x, y ∈ R, |y| < h} (i =−1). T = R/(2π Z). – ρ , d(U), a(U), J , D : see Section 2.3.1. U U U 1/2 1/2 AA CC h h –W = T × U, W ={(z,w) ∈ D(0, e ρ ) × D(0, e ρ ), −izw ∈ U}. h,U h,U U U −δ – e W = W −δ . h,U h−δ/2,e U – O (W): the set of σ -symmetric holomorphic function on W. – O (W ): see Notation 2.3. Symp (W ): see Notation 4.8. σ h,U h,U a,C a – δ = d (F; W ), δ = d(F; W ), O (F, G), O (F, G):Section 2.5. h,U h,U l – TC(A, B), (A, B)-twist condition: see Notation 2.6 and Deﬁnition 2.4. – G(f , W), L(f , W): see Notation 4.1. – = φ . For the canonical map f see (4.87)and (4.88). F F J∇ F – []· Y = Y ◦ − Y. See Section 4.7. – M (F),T F, R F: see Section 5.1. n N N –A"B = (A ∪ B) (A ∩ B). 3. A no-screening criterion on domains with holes 3.1. Harmonic measures. — Let U be a bounded open set of the complex plane with boundary ∂ U. We can deﬁne its Green function, g : U × U → R as follows: for any z ∈ U, −g(z,·) is the function equal to 0 on the boundary ∂ U of U, which is subharmonic on U, harmonic on U {z} and which behaves like log|z − w| when w ∈ Ugoes to z (this means that g(z,w) + log|z − w| stays bounded when w goes to z). The Green function g is thus nonnegative. We denote by ω : U × Bor(∂ U) →[0, 1] the harmonic measure U U of U (here Bor(∂ U) is the set of borelian subsets of ∂ U) deﬁned as follows: if z ∈ Uand I ∈ Bor(∂ U) (one can assume I is an arc for example if ∂ U is a union of circles) then the function ω (·, I) is the unique harmonic function deﬁned on U, having a continuous U 34 RAPHAËL KRIKORIAN extension to Uand such that ω (z, I) = 1if z ∈ Iand 0 if z ∈ ∂ U I. Poisson-Jensen formula (cf. [39]) asserts that for any subharmonic function u : U → C u(z) = u(w)dω (z,w) − g (z,w)u(w) U U ∂ U U where u is the usual Laplacian of u. In particular, if f is a holomorphic function on U, the application of this formula to u(z) = ln|f (z)| gives ln|f (z)|= ln|f (w)|dω (z,w) − g (z,w) U U ∂ U w:f (w)=0 and thus since g is nonnegative (3.66)ln|f (z)|≤ ln|f (w)|dω (z,w). ∂ U Though we shall not use it in this paper we mention the fact that the harmonic measure ω (z,·) can also be deﬁned in a probabilistic way by using Brownian motions: if W (t) is the value at time t of a Brownian motion issued from the point z (at time 0) and T is the stopping time adapted to the ﬁltration F of hitting I before ∂ U I, then z,I z ω (z, I) = E(1 (W (T )));hence U I z z,I (3.67)ln|f (z)|≤ E(ln|f (W (T ))|). z z,I This probabilistic interpretation is often useful in trying to get a hunch of the behavior of the harmonic measures. 3.2. Screening effect. — Assume now that |f |≤ 1onUand that |f | 1on some nonempty open subset B ⊂ U. Does this imply that f is small on “most” of U? Formula (3.66) applied to the domain U B in place of U yields for any z ∈ U B (3.68)ln|f (z)|≤ ω (z,∂ B) × lnf UB and answering the preceding question amounts to getting good estimates from below on the nonnegative function ω (z,∂ B). UB For example take U = D(0, 1) and B = D(0,σ ) (∂(U B) is then the union of the two circles of center 0 and radii σ and 1). It is easy to see that for |z|≤ 1/2 (3.69) ω (z,∂ B) = ln|z|/ ln σ ≥ ln(1/2)/ ln σ UB and the preceding formula (3.68) applied to the domain U Bshows that (3.70)lnf lnf . D(0,1/2) D(0,σ ) | ln σ| ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 35 If we assume for instance α β −N −N f ≤ e , e ≤ σ ≤ 1/10, 0 <β <α, N 1 this yields −β (3.71) ω (z,∂ D(0,σ )) N UD(0,σ ) α−β (3.72)lnf −N −1. D(0,1/2) Our aim in the next Section 3.3 will be to generalize to more general domains U, in particular to holed disks (3.73)U = D(0,ρ) D(c ,ρ ) ⊃ D(0,σ ) j j j=1 (j = 1,..., N, D(c ,ρ ) ⊂ D(0,ρ), ρ σ ), the bound from below (3.69) and its immedi- j j j ate consequence inequality (3.70). However, the spectral properties of the Laplacian (with Dirichlet boundary condi- tions for example) on a domain U obtained from removing disks from a simply connected domain ∈ R C (say the unit disk), and, in particular, the possibility of having useful estimates such as (3.71), (3.72), depend(s) on the number and the sizes of the holes of U. This fact, well known in Electrostatics under the name of screening effect,was mathemat- ically studied by Rauch and Taylor in [40] (see also [41]) where they highlight two dif- ferent regimes: on the one hand, if the sizes of the holes of U are (very) small compared with their number, the spectral properties of the Laplacian on U are very similar to that of ; on the other hand, if the holes are not so small and become dense (in some sense) in a region ⊂ (which can be of codimension 1 in ) the spectral properties of the Laplacian are similar to those of . In this latter case, the holes may act like a screen that prevents the propagation of the information “ln f −1on D(0,σ )” to the rest of the holed domain U. In Appendix C we illustrate this phenomenon on an example. We now give conditions under which this screening phenomenon is not effective. 3.3. The no-screening criterion. Proposition 3.1. — Let U be a domain U = D(0,ρ) ( D(z ,ε )), D(z ,ε ) ⊂ j j j j 1≤j≤N D(0,ρ) (ρ ∈]0, 1[)and let B ⊂ U, B = D(0,σ ). Assume that f ∈ O(U) satisﬁes f ≤ 1 and f ≤ m. ∂ B 36 RAPHAËL KRIKORIAN Then for any point z ∈ U := D(0,ρ) ( D(z , d )), 2ε < d < 1 j j j j 1≤j≤N ln(|z|/ρ) ln(d /2ρ) (3.74)ln|f (z)|≤ − ln m. ln(σ/ρ) ln(ε /ρ) j=1 Proof. — Replacing z/ρ by z, z /ρ by z , σ/ρ by σ , ε /ρ by ε and d /ρ by d ,we j j j j j j can reduce to the case ρ = 1. We then denote D = D(0, 1),D = D(z ,ε ),B = D(0,σ ). j j j By Poisson-Jensen formula (3.75) ln|f (z)|≤ ln|f (w)|dω (z,w) UB ∂(UB) ≤ ω (z,∂ B) ln m. UB We now compare ω (z,∂ B) with ω (z,∂ B). We observe that the function z → UB DB ω (z,∂ B) is the unique harmonic function deﬁned on U B which is 1 on ∂Band 0 UB on ∂ D ∪ ∂(D U); since ∂(U B) = ∂ B ∪ ∂ D ∪ ∂(D U) we deduce by the Maximum Principle that it takes its values in [0, 1]. Similarly, the function z → ω (z,∂ B) is the DB unique harmonic function deﬁned on D B which is 1 on ∂Band 0 on ∂ D, hence it takes also its values in [0, 1].So (3.76) v(·) := ω (·,∂ B) − ω (·,∂ B) UB DB is a harmonic function deﬁned on U B, −1 ≤ v ≤ 1, which is 0 on ∂ B ∪ ∂ D. For 1 ≤ j ≤ N, let v be the harmonic function deﬁned on D (B ∪ D ) which is 0 j j on ∂(D B) = ∂ D ∪ ∂Band −1on ∂ D ; by the Maximum Principle −1 ≤ v ≤ 0. j j Lemma 3.2. — The function v is harmonic on U B and on this set j=1 v ≤ v. j=1 Proof. — We notice that the function v is deﬁned and harmonic on D (B∪ j=1 N N D ) = U B. We want to compare v and v on the boundary ∂(U B) = j j j=1 j=1 ∂ D ∪ ∂ B ∪ ∂(D U).On ∂ D ∪ ∂ B the two functions v and v are equal (they are j=1 both equal to 0). To compare them on ∂(D U) we notice that ∂(D U) ⊂ ∂ D j=1 and since v =−1and for i = j , v ≤ 0 we have at each point z ∈ ∂(D U) which is j | ∂ D i N N in ∂ D , v (z)≤−1hence v ≤−1. But we have seen that −1 ≤ v ≤ 1 j i i | ∂(DU) i=1 i=1 on U B. We have thus proven that on ∂(U B) one has v ≤ v and we conclude j=1 the proof by the Maximum Principle. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 37 Because of the Maximum Principle, one has on D (B ∪ D ) ln|z − z|− ln 2 − ≤ v (z). ln ε Using Lemma 3.2 we hence get for z ∈ U, N N N ln|z − z|− ln 2 ln(d /2) j j v(z) ≥ v (z)≥− ≥− . ln ε ln ε j j j=1 j=1 j=1 On the other hand ln|z| ω (z, B) = , DB ln σ so that from (3.76) one has for z ∈ U ln|z| ln(d /2) ω (z, B) ≥ − . UB ln σ ln ε j=1 Finally since ln m ≤ 0, (3.75)gives that forany z ∈ U ln|z| ln(d /2) ln|f (z)|≤ − ln m. ln σ ln ε j=1 In particular, if for example ln(d /2) ln|z| ≤ (1/2) ln ε ln σ j=1 ln|z| then ln|f (z)|≤ (1/2) ln m, an inequality which is quite similar to (3.70). ln σ 3.4. Good triples. Deﬁnition 3.3. — Let U, U , U be three nonempty open sets of C such that, 1 2 U ⊂ U, U ⊂ U. 1 2 We say that the triple (U, U , U ) is A-good (A > 0)iffor any f ∈ O(U) such that sup |f |≤ 1, 1 2 one has lnf ≤ Alnf . U U 1 2 38 RAPHAËL KRIKORIAN FIG.8.—Atriple (U, U , U ) 1 2 Remark 3.1. — Notice that if there exists an open set U ⊂ U, U ⊂ U ,U ⊂ U 1 2 such that (U , U , U ) is A-good, then (U, U , U ) is also A-good. 1 2 1 2 Remark 3.2. — In general the fact (U, U , U ) is A-good does not imply that 1 2 (U, U , U ) is A -good with A and A comparable. For example, if U = D(0, 1),U = 2 1 1 D(1/4,σ ) U = D(3/4,σ ) with σ ,σ < 1/10, (U, U , U ) is C/| ln σ |-good while 1 2 2 1 2 1 2 2 (U, U , U ) is C/| ln σ |-good. 2 1 1 We denote by A(z; λ ,λ ),0 <λ <λ , the annulus D(z,λ ) D(z,λ ). 1 2 1 2 2 1 Here is an immediate corollary of Proposition 3.1: Corollary 3.4. — Assume that the assumptions of Proposition 3.1 hold with σ = ρ /2 −δ (b > 1). Then for all 1 ≤ i ≤ N such that d > 20ε and D(z , d ) ⊂ D(0, e ρ) (δ> 0), the i i i i triple U, A(z ; (d /10), d ), D(0,ρ /2) i i i is A-good with δ ln(d /20ρ) A = − . b| ln ρ| ln(ε /ρ) j=1 4. Symplectic diffeomorphisms on holed domains 4.1. Cartesian Coordinates (CC) and Action-Angle variables (AA). — We deﬁne on R := {(x, y), x, y ∈ R} (resp. T × R := {(θ , r), θ ∈ T, r ∈ R}) the canonical sym- CC∗ AA plectic structure (area) β := dx ∧ dy (resp. β := dθ ∧ dr). This spaceaswellas R R its symplectic structure can be complexiﬁed: the space C := {(x, y), x, y ∈ C} (resp. CC∗ T × C := {(θ , r), θ ∈ T , r ∈ C}) carries the symplectic structure β := dx∧ dy (resp. ∞ ∞ C ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 39 AA 2 CC∗ β := dθ ∧ dr) and the involution σ (resp. σ )deﬁnedin(2.42) preserves (C ,β ) 1 3 C C AA 2 CC∗ AA (resp. (T × C,β ))and ﬁxes (R ,β ) (resp. (T × R,β )). C R R When working in the elliptic ﬁxed point case, it will be more convenient to use other Cartesian coordinates. Let’s introduce the (holomorphic) complex change of coor- 2 2 dinates ϕ : C → C , ϕ : (x, y) → (z,w), 1 1 √ √ z = (x + iy) x = (z − iw) 2 2 (4.77) ⇐⇒ i −i √ √ w = (x − iy) y = (z + iw). 2 2 We see that (σ is as in (2.42)) with the notations of Section 2.1 ∗ −1 CC∗ CC dx ∧ dy = ϕ (dz ∧ dw), ϕ ◦ σ ◦ ϕ = σ,ϕ(W ) = W . 1 2 h,U h,U CC CC CC∗ CC We shall denote (M ,β ,σ ), resp. (M ,β ,σ ), (CC stands for Carte- 2 1 2 CC sian Coordinates) the space C endowed with the symplectic structure β := dz ∧ dw, CC∗ AA AA resp. β = dx ∧ dy, and the involution σ , resp. σ . Similarly, (M ,β ,σ ) (AA for 2 1 3 Action-Angle coordinates) is the space T × C endowed with the symplectic structure AA β := dθ ∧ dr and the involution σ . We shall use for short the generic notation (M,β,σ ) to denote either of the preceding sets endowed with their symplectic structure and invo- lution. We also use the notation M or (M) for M ∩ σ(M).The 2-form β restricted R R to M is still a symplectic form. We shall call the origin OinM , the set O ={(0, 0)} if R R 2 CC CC AA M = C = M or M and O = T×{0} if M = M = T × C. If W is a nonempty open set of M (resp. M ) we say that f ∈ Diff (W) (resp. C ∗ f ∈ Diff (W))is symplectic if it preserves the canonical symplectic form β : f β = β . O C We denote by Symp (W) (resp. Symp (W)) the set of such symplectic holomorphic 1 O (resp. C ) diffeomorphisms. If furthermore f ◦ σ = σ ◦ f we write f ∈ Symp (W). We shall say that a symplectic diffeomorphism f is exact symplectic if there exists a 1- form λ,the Liouville form,suchthat dλ = β and f λ − λ is exact: there exists a function Ssuch that f λ − λ = d S; S is called the generating function of f (w.r.t. λ). We then denote O C f ∈ Symp (W) (resp. f ∈ Symp (W)). In our case the relevant Liouville forms will be ex.,σ ex. (4.78) (AA) λ = rdθ, (CC) λ = (1/2)(wdz− zdw), (CC*) λ = (1/2)(xdy− ydx). Let W ⊂ Mbe σ -symmetric (σ(W) = W) and such that (W) := W ∩ σ(W) = W ∩ M R R is a nonempty open set of M . Then, if f ∈ Symp (W), its restriction f : M ⊃ R |(W) R ex.,σ (W) → f ((W) ) ⊂ M deﬁnes a real analytic (exact) symplectic diffeomorphism. If R R R S ⊂ Wis f -invariant (f (S) = S) the set (S) := S ∩ M is also left invariant by f . R R |(W) Notice that if U ⊂ C is a real symmetric open set such that U ∩ R = ∅ we have AA AA ⎪ (W ) = (T × U) = T × (U ∩ R) = W R h R ⎨ h,U 0,U∩R CC CC CC (W ) ={(z,w) ∈ W ,w = iz}= (W ) R R h,U 0,U∩R 0,U∩R + + 2 2 x +y CC∗ 2 CC∗ (W ) ={(x, y) ∈ R , ∈ U ∩ R }= W . R + h,U 0,U∩R 2 + 40 RAPHAËL KRIKORIAN In any case (W ) ={r ∈ U}∩ M ={r ∈ U ∩ R}∩ M . h,U R R R There are symplectic changes of coordinates ψ that allow to pass from the (z,w)- coordinates ((CC)-coordinates) to the (θ , r)-coordinates ((AA)-coordinates). They are de- 1/2 is 1/2 is/2 ﬁned as follows. The maps r → r , te → t e for t > 0and −π< s <π (resp. for t > 0and 0 < s < 2π ) deﬁne holomorphic functions on C R (resp. on C R ). We − + can thus deﬁne the biholomorphic diffeomorphisms T × (C R ) (θ , r) −→ (z,w)∈{(z,w) ∈ C , −izw/∈ R } ∞ ± ± (4.79) iπ/4 1/2 −iθ r =−izw z = e r e ⇐⇒ 1/2 (−izw) iπ/4 1/2 iθ iθ −iπ/4 w iπ/4 w = e r e e = e = e 1/2 (−izw) z which satisfy −1 dz ∧ dw = dθ ∧ dr and ψ ◦ σ ◦ ψ = σ . ± 2 3 Notice that if h > 0 ψ −izw ∈ D(0,ρ) R (4.80) T × (D(0,ρ) R ) −→ (z,w) ∈ C , h ± −2h 2h e < |z/w| < e hence with the notations of Section 2.1 CC AA (4.81)W ⊃ ψ (W ). h,UR h,UR ± ± CC AA 4.2. Symplectic vector ﬁelds. — If (M,β) = (M ,β) or (M ,β) and F ∈ O (M) we deﬁne the holomorphic symplectic vector ﬁeld X by i β = d F. If J is the matrix one has −10 X = J∇ F. t 1 We denote by φ the ﬂow at time t ∈ R of the vector ﬁeld J∇Fand = φ its J∇ F J∇ F time 1-map. It is a symplectic diffeomorphism. If G : M → R or C is another smooth observable we deﬁne the Poisson bracket of F and G by the formula {F, G}= β(X , X ) or equivalently F G {F, G}:=∇ F, J∇ G. One then has (G ◦ ) = L G={F, G}, [L , L ]= L . |t=0 J∇ F X X X F F G {F,G} dt If f is a symplectic diffeomorphism one has −1 −1 ∗ −1 (4.82) f ◦ ◦ f = , where f F = (f ) F = F ◦ f . F f F ∗ ∗ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 41 2 2 4.3. Integrable models. — We assume that (M,β,σ ) is (C , dx ∧ dy,σ ), (C , dz ∧ dw, σ ) or (T × C, dθ ∧ dr,σ ). In all these examples there exists a natural (Lagrangian) 2 ∞ 3 foliation given by the level lines of the observable r : M → C 2 2 x + y (4.83) r(x, y) = , r(z,w)=−izw, r(θ , r) = r, which has the property that for every m ∈ M, such that r(m) ∈ R,the map R t → φ (m) is 2π -periodic. In particular, for c ∈ R, the set {r = c}⊂ M is itself foliated by the J∇ r t 1 2π -periodic orbits of the ﬂow φ ; they are either points or homeomorphic to S .We J∇ r shall say that a symplectic diffeomorphism of M is integrable if it is symplectically conju- gated to a diffeomorphism that leaves globally invariant each level line of the preceding function r. It is not difﬁcult to see that a diffeomorphism satisfying the previous condition is of the form where H = ◦ r. Let U be a σ -symmetric holed domain of C and ∈ O (U). Then, −i∂(r) i∂(r) (CC) : (z,w) = (e z, e w), (4.84) (AA) : (θ , r) = (θ + ∂(r), r), −i∂(r) −i∂(r) (CC∗) : (x, y) = (#(e (x + iy)),$(e (x + iy))) and in any case (W ) ⊂ W , h = h−$(∂ ) . h,U U h,U On the other hand, since is σ -symmetric, one has whenever U is σ -symmetric, ((W ) ) = ((W ) ). h,U R h,U R Notice that in all cases is an integrable diffeomorphism of M. CC∗ 2 CC AA 4.4. KAM circles. — A circle of M (M equals M = R ,M ,M = T× R)is R R R R R any set of the form ({r = c}) = ({r = c)}∩ M , c ∈ R, of cardinal > 1(r is the observable R R of (4.83)). In the (AA) resp. (CC*) cases this set coincides with the usual circle T×{r = c} 2 2 2 resp. {(x, y) ∈ R ,(1/2)(x + y ) = r}; in the (CC) or (CC*) cases ({r = c}) is a circle if and only if c > 0 (it is empty if c < 0 and reduced to {(0, 0} if c = 0). C 1 Let W be an open subset of M and f ∈ Symp (W) aC symplectic diffeo- ex. morphism W → f (W). For example f could be the restriction on W = (W ) of h,U R f ∈ Symp (W ),W ⊂ M. A KAM-circle (or KAM-curve)for f is the image g(({r = h,U h,U ex.,σ c}) ) ⊂ Wof a circle ({r = c}) , c ∈ R,bya C symplectic diffeomorphism g : M → M R R R R ﬁxing the origin (g({r = 0} )={r = 0} )and such that R R −1 g ◦ f ◦ g = + O(r − c), ω ∈ R Q. 2πωr 42 RAPHAËL KRIKORIAN The set g(({r = c}) ) ⊂ Wis then f -invariant, homeomorphic to S and non homotopi- cally trivial in the following sense: in the (AA)-case it is homotopic to {r = 0} = T×{0} and in the (CC) or (CC*) case it has degree ±1 w.r.t. to the origin {r = 0} ={(0, 0)}. Moreover, the restriction of f on g(({r = c}) ) ⊂ W is conjugated to a rotation on a circle with frequency ω ∈ R. Notation 4.1. — We denote by G(f , W) the set of f -invariant KAM-circles γ ⊂ (W) and by L(f , W) ⊂ (W) their union: L(f , W) = γ . γ∈G (f ,W) Remark 4.1. —Let g, f , f , f : M → M be C symplectic diffeomorphisms 1 2 R R where g({r = 0} )={r = 0} . Then, R R (1) If A ⊂ B ⊂ (M) ,then L(f , A) ⊂ L(f , B). (2) If f , f coincide on a set A, L(f , A) = L(f , A). 1 2 1 2 (3) For any set A ⊂ M −1 (4.85) g(L(f , A)) = L(g ◦ f ◦ g , g(A)). −1 (4) If g ◦ f ◦ g and f coincide on a set A one has 1 2 (4.86) L(f , g(A)) = g(L(f , A)). 1 2 Deﬁnition 4.2. — If A ⊂ C we deﬁne W ={r ∈ A}∩ M ={r ∈ A ∩ R}∩ M . A R R Let us now state a criterion that ensures the existence of KAM-circles. Assume that there exist ∅ = L ⊂ A = I I ⊂ A ⊂ R, j∈J where L is compact and A is of the form I I where I ⊂ R is an interval and the I j j j∈J are pairwise disjoint intervals. C 1 Proposition 4.3. — Let f ∈ Symp (W ) and suppose that there exist ∈ C (R) and a 1 −1 C symplectic diffeomorphism g : M → M ﬁxing the origin, g − id 1 ≤ C (C depends only R R C on M), such that −1 on W g ◦ f ◦ g = and g(W ) ⊂ W . L (r) L A 1/2 Then, if |I | ≤ 1, one has j∈J 1/2 Leb (W L(f , W )) ≤ C × (Leb (A L)+g − id ). M A R 0 R A Proof. — Since W = L( , W ) one has from (4.86) g(W )= g(L( , W ) = L (r) L L (r) L L(f , g(W )) and since g(W ) ⊂ W one has g(W ) ⊂ L(f , W ). On the other hand L L A L A ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 43 if we deﬁne E by W = W ∪ E, one has g(W ) = g(W ) ∪ g(E) and thus g(W ) ⊂ A L A L A L(f , W ) ∪ g(E). We therefore have Leb (W L(f , W )) Leb (g(E)) + Leb (W " g(W )). M A A M M A A R R R 1/2 Since A = I I and |I | ≤ 1, Lemma J.1 from the Appendix yields j j j∈J j∈J 1/2 Leb (W " g(W ))≤g − id and since Leb (g(E)) = Leb (E) we get the con- M A A 0 M M R R R clusion. 4.5. Generating functions. — Let h > 0, U ⊂ C be a real symmetric holed domain AA CC and W and W the domains deﬁned in (2.47)and (2.46) h,U h,U AA W = T × U h,U 1/2 1/2 CC h h W ={(z,w) ∈ D(0, e ρ ) × D(0, e ρ ), r := −izw ∈ U}. h,U U U We shall associate to each F ∈ O (W ) small enough a real symmetric holomorphic σ h,U symplectic diffeomorphism f of W which is exact with respect to the respective Liouville F h,U forms as deﬁned in (4.78)). It is deﬁned as follows: in the (AA)-case ϕ = θ + ∂ F(θ , R) (4.87) f (θ , r) = (ϕ, R) ⇐⇒ r = R + ∂ F(θ , R) and in the (CC)-case z = z + ∂ F(z, w) (4.88) f (z,w) = ( z, w) ⇐⇒ w = w + ∂ F(z, w) . Lemma 4.4. — There exists a constant C such that if F ∈ O (W ) and 0 <δ < h satisfy σ h,U −2 (4.89) C(δ d(W )) F < 1, h,U W h,U the map f deﬁned by (4.87), (4.88) is a real symmetric holomorphic exact symplectic diffeomorphism −δ from e W onto its image and h,U −2δ −δ (4.90) e W ⊂ f (e W ) ⊂ W . h,U h,U h,U We shall call f the generating map of F.Moreover −1 (4.91) f = f . −F+O(D FDF) Proof. — See Appendix A.1. 44 RAPHAËL KRIKORIAN Remark 4.2. — The symplectic change of coordinates ψ introduced in Sec- CC tion 4.1 preserves exact symplecticity: if f is exact symplectic the same is true for AA −1 CC iπ/4 1/2 −iθ iπ/4 1/2 iθ f = ψ ◦ f ◦ ψ . Indeed, if ψ (θ , r) = (z,w), z = e r e , w = e r e ,one ± ± computes the Liouville form (1/2)(wdz − zdw) = rdθ . Conversely, if a diffeomorphism (θ , r) → (ϕ, R) is exact symplectic and close enough to the identity, it admits this type of parametrization. More precisely: Lemma 4.5. — Let f ∈ Symp (W ) be an exact symplectic diffeomorphism close enough h,U ex.σ −δ to the identity. Then, if δ = d(f − id, W ) (recall the notation (2.64)) there exists F ∈ O (e W ) h,U σ h,U −δ such that on e W one has h,U f = f , F = O(Df − id) = O (f − id). F 1 This F is unique up to the addition of a constant. Conversely, given F ∈ O(W ) one has h,U (4.92) f = ◦ f = id + J∇ F + O(D FDF). F F O (F) Proof. — See Appendix A.2. The composition of two exact symplectic maps is again exact symplectic and more precisely −δ Lemma 4.6. — Let F, G ∈ O(W ).If δ = d(F, G; W ) then on e W , h,U h,U h,U (4.93) f ◦ f = f F G F+G+O(DF DG ) h,U h,U (4.94) f = f ◦ f = f ◦ f . F+G F+DF O (G) G F G+DG O (F) h,U 1 h,U 1 In the Action-Angle case, if depends only on the variable r then = f and (4.95) ◦ f = f F +F Proof. — See the Appendix, Section A.3. 4.6. Parametrization. — We shall parametrize perturbations of integrable symplectic diffeomorphisms deﬁned on a domain W by h,U f = ◦ f (r) F where ∈ O (U) and F ∈ O (W ).Notethatif f = id + O (z,w) or f (θ , r) = id + σ σ h,U F (O(r), O(r )) then: 3 2 Case (CC) F(z,w) = O (z,w), Case (AA) F(θ , r) = O(r ). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 45 4.7. Transformation by conjugation. — We now deﬁne (4.96) []· Y = Y ◦ − Y. Note that (AA)-case if Y = Y(θ , r), ([]· Y)(θ , r) = Y(θ + ∂(r), r) − Y(θ , r); (4.97) (CC)-case if Y = Y(z,w), −i∂(r) i∂(r) ([]· Y)(z,w) = Y(e z, e w) − Y(z,w). If W = W is a holed domain and δ> 0 we introduce the notation h,U W = W := W ∪ (W ). h,U h,U h,U h,U The main result of this section is the following: Proposition 4.7. — Let ∈ O (U), F ∈ O (W ), Y ∈ O (W ).Then, if δ = σ σ h,U σ h,U −δ d(F, W ) ∩ d(Y, W ) there exists F ∈ O (e W ) such that h,U σ h,U h,U −δ −1 [e W ] f ◦ ( ◦ f ) ◦ f = ◦ f h,U Y F F (see the notation (2.44)) and F = F+[]· Y+DF O (Y) W 1 = F+[]· Y + O (Y, F). Proof. — See the Appendix, Section A.4. Remark 4.3. — A direct computation shows that if (r) = 2πω r + O(r ) and k k Case (CC) F(z,w) = O (z,w) and Y(z,w) = O (z,w), k k Case (AA) F(θ , r) = O(r ) and Y(θ , r) = O(r ) then 2k−2 2k−1 Case (CC) F(z,w) = O (z,w), Case (AA) F(θ , r) = O (r). 4.8. Symplectic Whitney extensions. — Let U ⊂ C be a real symmetric holed domain Wh 2 W ⊂ M, F ∈ O (W ) and F : M → C be a σ -symmetric C Whitney extension of h,U σ h,U (F, W ) (cf. Section 2.4). There exists a constant C > 0 (depending only on M) such that h,U Wh −1 1 2 Wh if F < C , Equations (4.87), (4.88)deﬁne a C -diffeomorphism f : M → M C (M) F such that −1 −1 Wh (4.98)max(f Wh − id 1 ,f − id 1 ) ≤ C F 2 . F C (M) Wh C (M) C (M) F 46 RAPHAËL KRIKORIAN −1 −1 1 −δ −δ Note that f Wh and f are C σ -symmetric extensions of (f , e W ) and (f , e W ) F Wh F h,U h,U for any δ satisfying (4.89), cf. Lemma 4.4. Wh Wh In general, the diffeomorphism f is not symplectic on M but since F takes real values on M , f Wh : M → M is an exact symplectic diffeomorphism of M . R F R R R Notation 4.8. — We shall denote by Symp (W ),resp. Symp (W ), the set of C h,U h,U σ ex.,σ O O σ -symmetric diffeomorphisms M → M that are in Symp (W ),resp. Symp (W ), (hence h,U h,U σ ex,s holomorphic on W ) and symplectic, resp. exact symplectic, when restricted to M → M . h,U R R 5. Cohomological equations and conjugations Our aim in this section is to provide a uniﬁed treatment, both in the (AA) and (CC) cases, of the resolution of the (co)homological equations (Proposition 5.3)involvedinthe Fundamental conjugation step (Proposition 5.5) that we shall use to construct all our dif- ferent Normal Forms (for instance the approximate Birkhoff Normal Form of Section 6, the KAM Normal Forms of Section 7 and the resonant Normal Form of Appendix G). 5.1. Fourier coefﬁcients and their generalization. — In this section we assume that either: – Case (CC): (M,β) = (C × C, dz ∧ dw) and we denote by r(z,w)=−izw –or, Case (AA): (M,β) = (T × C, dθ ∧ dr) and we denote by r : (θ , r) → r. In both cases the ﬂow t → φ is 2π -periodic w.r.t. t ∈ R (cf. (4.84)). J∇ r Let U be a connected open set of C and F ∈ O(W ).For any m ∈ W and any h,U h,U t ∈ R, φ (m) ∈ W : h,U J∇ r t −it it (CC) : φ (z,w) = (e z, e w), J∇ r (AA) : φ (θ , r) = (θ + t, r). J∇ r We can hence deﬁne t → F(φ (m)) which is a 2π -periodic function R → C and for J∇ r n ∈ Z we introduce its n-th Fourier coefﬁcient M (F)(m): 2π −int t (5.99) M (F)(m) = e F ◦ φ (m)dt J∇ r 2π t int (5.100) F(φ (m)) = M (F)(m)e . J∇ r n∈Z Thedependenceof M (F)(m) is holomorphic in m andwehavethusdeﬁned M (F) ∈ n n O(W ). We observe that h,U 2π/n (5.101) M (F) ◦ φ = M (F) n n J∇ r and ∀ t ∈ R, M (F) ◦ φ = M (F). 0 0 J∇ r ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 47 5.1.1. Case (CC). — One has t −it it (5.102) φ (z,w) = (e z, e w) J∇ r and if F = F(z,w),(5.99) becomes 2π −int −it it M (F)(z,w) = e F(e z, e w)dt. 2π k l If furthermore F(z,w) = F z w is converging on some polydisk D(0,μ) × k,l (k,l)∈N D(0,μ) one has k l (5.103) M (F)(z,w) = F z w n k,l (k,l)∈N l−k=n ∗ p hence, if for some p ∈ N ,F(z,w) = O (z,w),thenfor any n ∈ N, M (F)(z,w) = O (z,w). 5.1.2. (AA) Case. — In that case φ (θ , r) = (θ + t, r) J∇ r and if F = F(θ , r) we deﬁne 2π −1 −int M (F)(θ , r) = (2π) e F(θ + t, r)dt inθ = F(n, r)e where 2π −1 −inθ F(n, r) = (2π) e F(θ , r)dθ is the n-th Fourier coefﬁcient of F(·, r). Notice that though F is only deﬁned on T × U, M (F) is deﬁned in T × U. n ∞ Remark 5.1. — We see from (5.99)thatiffor some p > 0, F = O (r) (which means p p that for any m ∈ W one has |F(m)|≤ C|r(m)| for some C > 0) then M F = O (r) for h,U n any n ∈ N. 2π/n Remark 5.2. — Using the fact that M (F) ◦ φ = M (F) one can show that n n J∇ r 2π/n 2π/n f ◦ φ = φ ◦ f both in the (AA) and (CC) Case. F F J∇ r J∇ r 2π/n 36 −2π i/n 2π i/n For example in the (CC)-case, since φ (z,w) = (e z, e w), the condition on F implies J∇ r −2π i/n 2π i/n F(e z, e w) = F(z, w) and the conclusion follows from (4.88). 48 RAPHAËL KRIKORIAN 5.1.3. For m of M (F). Lemma 5.1. — If F ∈ O (W ) there exists M(F) ∈ O (U) such that σ h,U σ (5.104) M (F) = M(F) ◦ r, M(F) ≤F . 0 U h,U Moreover (5.105) f = ◦ f . M(F) M(F) O (F) Proof. — By deﬁnition of M (F) we see that for every t ∈ R M (F) ◦ φ = M (F). 0 0 J∇ r Lemma D.1 of the Appendix provides us with M(F) ∈ O (U) such that M (F) = σ 0 M(F) ◦ r.Wejusthavetoprove (5.105) in the (CC) case. If ( z, w) = f (z,w) one has M(F) −1 z = (1 + ∂(M(F))(zw) )z, w = (1 + w∂ (M(F))(zw) ) w and since w( z,w) = w + O(F) we get −∂(M(F))(zw) ∂(M(F))(zw) ( z, w) = (e z, e w) + O (F). 5.1.4. Decay of the M (F).— We observe that –in Case (AA),for m = (θ , r) ﬁxed in W , the function h,U T → C h−|$θ| t → F(φ (m)) = F(θ + t, r) J∇ r is well deﬁned and holomorphic; CC –in Case (CC),for (z,w) ∈ W ﬁxed (recall (5.102) and the deﬁnition (2.46)of W ), h,U h,U the function h 1/2 h 1/2 R + i]− ln(e ρ /|w|), ln(e ρ /|z|)[→ C (5.106) t −it it t → F(φ (m)) = F(e z, e w) J∇ r (with ρ = sup{|r|, r ∈ U}) is also a well deﬁned 2π Z-periodic holomorphic func- CC h−δ 1/2 tion. Furthermore, if m = (z,w) ∈ W one has max(|z|,|w|) ≤ e ρ thus h−δ,U h 1/2 h 1/2 min(ln(e ρ /|w|), ln(e ρ /|z|)) ≥ δ. Hence, in any case, for m ∈ W the function t → F◦ φ (m) is 2π -periodic, holomor- h−δ,U J∇ r phic on T and bounded in module by F . The Fourier coefﬁcients M (F)(m) of δ W n h,U the function t → F ◦ φ (m) J∇ r t int (5.107)F ◦ φ = e M (F) J∇ r n∈Z ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 49 thus satisfy 1/2 (5.108) |M (F)(m)| ≤F n W h,U n∈Z and in fact (cf. for example [44]) 1/2 2|n|δ 2 1/2 (5.109) e |M (F)(m)| ≤ 2 F . n W h,U n∈Z 5.1.5. Truncations operators. — Let us deﬁne for N ∈ N∪{∞} , T F = M (F), R F = F − T F. N n N N |n|<N Lemma 5.2. — If F ∈ O(W ) one has h,U (5.110) on W , F = M (F) h,U n n∈Z −|n|δ (5.111) M (F) e F , n W W h−δ,U h,U −1 −Nδ (5.112) R F δ e F , N W W h−δ,U h,U −1 −Nδ (5.113) T F F (if δ e ≤ 1). N W W h−δ,U h,U Furthermore, if for some p > 0, F = O (r) then (5.114)R F = O (r); in the (CC Case), if F ∈ O(W ) ∩ O (z,w), one has h,U (5.115) (R F)(z,w) = O (z,w). Proof. — Inequality (5.111) is a straightforward consequence of (5.109). Equality (5.110) comes from taking t = 0in (5.100). (5.112) is a consequence of (5.111)and (5.113) is clear from (5.112). Inequalities (5.114)and (5.115) are consequences respectively of Remark 5.1 and of identity (5.103). 5.2. Solution of the truncated cohomological equation. — We assume that 0 <ρ ≤ 1and that U is a σ -symmetric open connected set of D. We recall that we have deﬁned in (4.96)(cf. Proposition 4.7)for any ∈ O(U) and Y ∈ O(W ) h,U []· Y = Y ◦ − Y. (r) The main Proposition is the following: 50 RAPHAËL KRIKORIAN Proposition 5.3. — Let τ ≥ 0, ∈ O (U), K > 0, N ∈ N ∪{∞} be such that one has on U ∗ −1 −τ (5.116) ∀ (k, l) ∈ N × Z, 1 ≤ k < N %⇒ |k ∂(·) − l|≥ K |k| . 2π Then, for any F ∈ O (W ), there exists Y ∈ O (W ) such that, on W , one has M (Y) = 0, σ h,U σ h,U 0 h,U M (Y) = 0 for |k|≥ N and (5.117)T F − M (F)=[]· Y. N 0 This Y satisﬁes for any 0 <δ < h −(1+τ) τ+1 (5.118) Y Kmin(δ , N )F . h,U h−δ,U Moreover, if we assume in addition that is of the form (r) = 2πω r, ω ∈ R,thenone canimprove 0 0 the exponent in (5.118): −τ τ (5.119) Y Kmin(δ , N )F . h,U h−δ,U Proof. — We observe that both in Case (AA) or Case (CC) one has on W ∩{r ∈ R} h,U (cf. (4.84)) ∂(r) = φ . (r) J∇ r Hence, if G is a function in O(W ) one has on W ∩{r ∈ R} h,U h,U 2π t+∂(r) −int M (G) ◦ = e G ◦ φ dt n (r) J∇ r 2π 2π in∂(r) −int t = e e G ◦ φ dt J∇ r 2π in∂(r) = e M (G) and since M (G) ∈ O(W ), the left hand side of the preceding equations can be holo- n h,U morphically extended to a function in O(W ).Wethenhavein O(W ) h,U h,U in∂(r) []· M (G) = (e − 1)M (G). n n in∂(r) −1 −τ Note that from Lemma M.1 one has for r ∈ U, |e − 1|≥ K |n| .Ifwedeﬁne Y by (5.120)Y = M (F) in∂(r) e − 1 0<|n|<N ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 51 we have from Lemma 5.2 τ −|n|δ Y K |n| e F W h,U h−δ,U 1≤|n|<N −(1+τ) τ+1 min(Kδ , KN )F h,U and []· Y = Y ◦ − Y = T F − M (F). N 0 This last formula shows that if we deﬁne Yon (W ) by Y ◦ = T F − (r) h−δ,U (r) N M (F) + Y the functions Y and Y coincide on (W ) ∩ W and thus Y can 0 (r) h−δ,U h−δ,U be holomorphically extended to (W ) ∪ W =: W and (r) h−δ,U h−δ,U h−δ,U −(1+τ) τ+1 Y min(Kδ , KN )F , h,U h−δ,U which is (5.118). The fact that M (Y) = 0 and its uniqueness (under the condition M (Y) = 0) 0 0 comes again from Lemma 5.2. Finally, the σ -symmetry of Y on W is clear. h,U −1 To conclude the proof of the Proposition we just have to check that if (2π) (r) ≡ ω ∈ R satisﬁes (5.116)then(5.119) holds. This is a result due to Rüssmann [45]thatwe now recall for completeness. In fact, Rüssmann proves that if D = min|nω − l|, D = min D n 0 j l∈Z 1≤j≤n one has −2 2 ∗ −2 (5.121) D ≤ (π /3)(D ) . n N n=1 2π inω From Lemma M.1 one has |e − 1|≥ 4min |nω − l|= 4D . Thus, if we apply l∈Z 0 n Cauchy-Schwarz inequality to (5.120)wehavefor ν = 0or ν = δ 1/2 1/2 −2|n|ν −2 2|n|ν 2 Y e D e M (F) W n h−δ,U n W h−δ,U 1≤|n|<N 1≤|n|<N 1/2 −2|n|ν −2 e D F (cf. (5.108), (5.109)). n h,U 1≤|n|<N –If N < ∞,wetake ν = 0and (5.121)gives ∗ −1 Y (D ) F W h,U h−δ,U N KN F . h,U 52 RAPHAËL KRIKORIAN –If N =∞ we take ν = δ. Taking into account (5.121), we perform an Abel summa- −2|n|ν −2 −2|n|ν −2 tion (discrete integration by part) on the sums e D , e D : 1≤n<N n 1≤−n<N n this yields 1/2 −2|n|δ −2(|n|+1)δ ∗ −2 Y (e − e (D ) F W h,U h−δ,U n 1≤|n|<∞ 1/2 −2|n|δ 2 2τ δe K |n| F h,U 1≤|n|<∞ −τ Kδ F . h,U Remark 5.3. —If in Proposition 5.3 U = D(0,ρ) is a disk centered at 0 and ilθ k (AA)-case F(θ , r) = F (l)e r ⎪ k k∈N l∈Z k l (CC)-case F(z,w) = F z w ⎪ k,l (k,l)∈N one has the more explicit expressions F (l) ilθ k (AA)-case Y(θ , r) = e r il∂(r) ⎨ e − 1 k∈N l∈Z (5.122) k,l k l (CC)-case Y(z,w) = z w . ⎪ i(l−k)∂ (r) ⎪ e − 1 (k,l)∈N l =k In particular, if (CC)-case F(z,w) = O (z,w) (r) = 2πω r and (AA)-case F(θ , r) = O (r) then Y satisﬁes also (see the remarks at the end of Sections 5.1.1 and 5.1.2) (CC)-case Y(z,w) = O (z,w) (AA)-case Y(θ , r) = O (r). 5.3. Fundamental conjugation step. — We begin by the following consequence of Proposition 4.7. Let U be a holed domain, h > 0. Lemma 5.4. — There exists a ≥ 2 and C > 0 such that if ∈ O (U), F ∈ O (W ), σ σ h,U Y ∈ O (W ) and δ = d(F, Y; W ), δ> 0 satisﬁes σ h,U h,U −a −a (5.123)C d(W ) δ F ≤ 1 h,U W h,U ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 53 −δ then one has on e W (cf. Lemma 5.1 for the deﬁnition of M(F)) h,U −1 (5.124) f ◦ ◦ f ◦ f = ◦ f (a) . Y F +M(F) ˙ F−M (F)+[+M(F)]·Y+O (Y,F) Proof. — We ﬁrst observe that since F = F − M (F) + M (F),wehaveby(4.94) 0 0 and Lemma 5.1 f = f ◦ f F M (F) F−M (F)+O (F) 0 0 2 = ◦ f ◦ f M(F) O (F) F−M (F)+O (F) 2 0 2 = ◦ f M(F) F−M (F)+O (F) 0 2 and thus ◦ f = ◦ f . F +M(F) F−M (F)+O (F) 0 2 Now we use Proposition 4.7 and make explicit the notations d and O:for some a > 0that we can choose ≥ 2and some C > 0, if (5.123)issatisﬁed, onehas −1 (5.125) f ◦ ◦ f ◦ f Y +M(F) F−M (F)+O (F) 0 2 Y (a) = ◦ f . +M(F) F−M (F)+[+M(F)]·Y+O (Y,F) Proposition 5.5. — Let a = a + 4 (a from Lemma 5.4). There exists C > 0 such that the following holds. Let U be a holed domain, ∈ O (U),and F ∈ O (W ). Assume that there exists σ σ h,U a holed domain V ⊂ U, N ∈ N ∪{∞} and K > 0 such that on V the following non-resonance condition (cf. (5.116)) is satisﬁed: ∗ −1 −τ (5.126) ∀ (k, l) ∈ N × Z, 1 ≤ k < N %⇒ |k ∂(·) − l|≥ K |k| 2π −1 −1 −δ and assume that CN <δ < min(h, C ) is such that e W is not empty and h,V −1 −(a +τ) (5.127) (δ d(W )) KF < C . h,V h,U Then there exists Y ∈ O(W ) solution on W of the cohomological equation (cf. (5.117), (5.118)): h,V h,V −(1+τ) (5.128)T F − M (F)=−[]· Y, Y −δ/2 Kδ F N 0 W e W h,U h,V −δ −δ and ∈ O(e W ), F ∈ O (W ) such that one has on e W h,V σ h,U h,V −1 f ◦ ◦ f ◦ f = ◦ f , = + M(F) Y (r) F (r) F (5.129) −(a +τ) 2 −Nδ/2 F 3 −δ ≤ K(δ d(W )) F + e F . C (e W ) h,V h,U h,V h,U 54 RAPHAËL KRIKORIAN Proof. — We apply Proposition 5.3 to obtain some Y satisfying (5.128)and we apply Lemma 5.4 with δ equal to δ/2. Since (cf. (4.97)) [ + M(F)]· Y =[]· Y + (2) O(|∇ Y||∇(M(F))|)=[]· Y+ O (Y, F), we get using []· Y+ F− M (F) = R F(cf. 0 N (5.117)), −δ/2 −1 e W , f ◦ ◦ f ◦ f =: ◦ f h,V Y F with (5.130) = + M(F) (a) (5.131) F = R F + O (Y, F). (a) The deﬁnition of the symbol O ,(5.112)and (5.128) show that there exists a universal positive constant Csuch that if (5.127) is satisﬁed one has −(1+τ) −1 −1 a 2 −1 −Nδ/2 (5.132) F −δ/2 Kδ (δ d(W ) ) F + δ e F . e W h,V h,U h,V h,U Inequalities (5.129), comes from (5.132) and Cauchy’s inequality (2.53)ofSection 2.3.4 −δ −δ/2 −δ −δ/2 (applied with e W and e W in place of e W and W ) because d(e W ) ≥ h,V h,V h,U h,U h,V (1/2) d(W ) if δ< 1/10 (which is the case if C is large enough). h,V 6. Birkhoff Normal Forms 6.1. Formal Normal Forms. — We recall in this subsection the classical results on (formal) Birkhoff Normal Forms. For more details on the related formal aspects we refer to Appendix E. We also explain how Pérez-Marco’s dichotomy extends to the diffeomor- phism case (in particular in the (AA)-case). 6.1.1. BNF near a non-resonant elliptic ﬁxed point ((CC) case). — Let f : (R , 0) → (R , 0) be a real analytic symplectic diffeomorphism of the form f (x, y) = Df (0, 0) · (x, y) + O (x, y) where cos(2πω ) − sin(2πω ) 0 0 Df (0, 0) = = 2πω r sin(2πω ) cos(2πω ) 0 0 with ω ∈ R Q. 2 2 If ϕ : C → C is the change of coordinates ϕ(x, y) = (z,w) deﬁned in (4.77) −1 the diffeomorphism f := ϕ ◦ f ◦ ϕ is exact symplectic and of the form f (z,w) = (z,w) + O (z,w) where r(z,w)=−izw 2πω r −2π iω 2π iω 0 0 (z,w) = (e z, e w). 2πω r 0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 55 From Lemma 4.5 we have the representation f = ◦ f , F = O (z,w) 2πω r F for some F ∈ O (D(0,μ) ), μ> 0. We then have the following classical proposition that establishes the existence of Birkhoff Normal Forms to arbitrarily high order. Proposition 6.1. — Let ω ∈ R Q. Then, for any N ≥ 3 there exist σ -symmetric holomor- 2 2 phic maps : (C, 0) → C, Z , F : (C , 0) → C such that on a neighborhood of 0 ∈ C one N N N has (r =−izw) −1 ⎪ f ◦ ( ◦ f ) ◦ f = ◦ f Z 2πω r F F ⎨ N 0 Z N N 2(N+1) 3 (6.133) F (z,w) = O (z,w), Z (z,w) = O (z,w), N N (r) = 2πω r + O (r). N 0 Remark 6.1. — The sequences (Z ) and ( ) converge respectively in C[[z,w]] N N N N and in R[[r]].If Z ∈ C[[z,w]] and ∈ R[[r]] are their respective limits one has in ∞ ∞ C[[z,w]] the formal identity −1 f ◦ ( ◦ f ) ◦ f = Z 2πω r F ∞ 0 Z ∞ (6.134) 3 2 Z (z,w) = O (z,w), (r) = 2πω r + O (r). ∞ ∞ 0 Conversely, (6.134)deﬁnes uniquely; is the Birkhoff Normal Form BNF(f ) of f ∞ ∞ (and BNF(f ) of f ). In particular, BNF(f ) is invariant by (formal) symplectic conjugacies which are tangent to the identity. Remark 6.2. —If f = ◦ f with = (r) = 2πω r + O(r ) and F(z,w) = F 0 2(N+1) O (z,w) then N+1 (6.135)BNF(f )(r) = (r) + O (r). 6.1.2. BNF near a KAM circle (Action-Angle case). — Let f : (T × R, T ×{0}) → (T × R, T ×{0}) be a real analytic symplectic diffeomorphism of the form f (θ , r) = (θ + 2πω , r) + (O(r), O(r )).Wenoticethat : (θ , r) → (θ + 2πω , r).Wecan 0 2πω r 0 thus write f under the form (h,ρ > 0) 2h 2 f = ◦ f , F ∈ O (e (T × D(0, ρ))), F = O (r). 2πω r F σ h The normalizing map Z is unique up to composition on the left by a formal generalized symplectic rotation , ∞ A A ∈ R[[r]]. 56 RAPHAËL KRIKORIAN Proposition 6.2. — Let ω ∈ R be Diophantine. Then, for any N ≥ 3 there exist real analytic maps : (R, 0) → R, Z , F : (T × R, T×{0}) → R such that N N N −1 f ◦ ( ◦ f ) ◦ f = ◦ f Z 2πω r F F N 0 Z N N (6.136) N+1 2 2 F (θ , r) = O (r), Z (θ , r) = O (r), (r) = 2πω r + O (r). N N N 0 ω ω ω Remark 6.3. —Let C (T)[[r]] (where C (T) = C (T)) be the set of formal h>0 h power series n ω (6.137)F(θ , r) = F (θ )r , F ∈ C (T) for all n ∈ N. n n n∈N The sequence (Z ) converges in C (T)[[r]] and the sequence ( ) converges in N N N N R[[r]].If Z ∈ C (T)[[r]] and ∈ R[[r]] are their respective limits one has in ∞ ∞ C (T)[[r]] the formal identity −1 f ◦ ( ◦ f ) ◦ f = Z 2πω r F ∞ 0 Z ∞ (6.138) 2 2 Z (θ , r) = O (r), (r) = 2πω r + O (r). ∞ ∞ 0 Conversely, (6.138)deﬁnes uniquely; is the Birkhoff Normal Form BNF(f ) ∞ ∞ of f . In particular, BNF(f ) is invariant by (formal) symplectic conjugacies which are of the form id + (O(r), O(r )). Remark 6.4. —If f = ◦ f with = (r) = 2πω r + O(r ) and F(θ , r) = F 0 N+1 O (r) then N+1 (6.139)BNF(f )(r) = (r) + O (r). Remark 6.5. — The reason why we impose a Diophantine condition on ω in the statement of Proposition 6.2 is the following. The existence of the formal Birkhoff Nor- mal Form (6.136) derives from an inductive procedure where at each step n ∈ N one con- ω ω structs a formal conjugation f with Y ∈ C (T)[[r]] that conjugates f (F ∈ C (T)[[r]], Y n F n n n n ω n+1 F = O (r))to f (F ∈ C (T)[[r]],F (θ , r) = O (r)). To perform this conjugation n F n+1 n+1 n+1 step one has to solve a cohomological equation F (θ , r) = Y (θ + 2πω , r) − Y (θ , r) + n n 0 n 2π F (ϕ, r)dϕ where r is a formal variable but θ lies on T (see Lemma E.7). This equation is classically solved by passing to Fourier coefﬁcients (see for example [13]) but it involves small denominators that can be dealt with if ω satisﬁes an arithmetic condition, for example a Diophantine one (weaker conditions such as Bruno condition or even ln q = o(q ) n+1 n will also be ﬁne ). The normalizing map Z is unique up to composition on the left by a formal integrable twist of the form , ∞ A A ∈ R[[r]]. As usual p /q are the convergents of ω . n n 0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 57 6.2. Pérez-Marco’s Dichotomy. — We now discuss the extension of Pérez-Marco’s Dichotomy, Theorem 1.3, to the diffeomorphism setting. The ﬁrst part of Pérez-Marco’s argument in [36], translated in our (CC)- setting, is based on the fact that the coefﬁcients of the Birkhoff Normal Form B(r) = n n b (F)r = b (F)(−izw) of ◦ f depend polynomially on the coef- d d n n 2πω,r F n∈N n∈N k l ﬁcients of F(z,w) = F z w . More precisely, if we denote by [F] , j ≥ 3, d d k,l j (k,l)∈N ×N k l the homogeneous part of F of degree j , [F] = F z w , then the coefﬁcients j k,l |k|+|l|=j of the homogeneous part of degree 2j , [B ◦ r] = b (F)(−izw) of B ◦ r,are 2j k |k|=j polynomials of degree 2j − 2 in the coefﬁcients of [F] ,...[F] . As a consequence, 3 j if (z,w) → F(z,w), (z,w) → G(z,w) are two σ -symmetric holomorphic functions 3 3 such that F(z,w) = O (z,w),G(z,w) = O (z,w),thenfor any n ≥ 3, the maps t → b (tF+ (1− t)G) are polynomials of degree ≤ 2|n|− 2. The second argument in [36] is then to use results from potential theory (in particular the Bernstein-Walsh Lemma ) applied to the family of polynomials t → b (tF + (1 − t)G) that have a degree which behaves linearly in n. To check that the arguments of [36] adapt to the diffeomorphism case it is hence enough to check that t → b (tF + (1 − t)G) are polynomials of degree ≤ 2(|n|− 1): Lemma 6.3. — If F, G are σ -symmetric holomorphic maps F, G = O (z,w) in the (CC)- 2 d case (resp. F, G = O (r) in the (AA)-case) then, for every n ∈ N , |n|≥ 2,t → b (tF + (1 − t)G) is a polynomial of degree ≤ 2(|n|− 1) (resp. ≤|n|− 1). Proof. — We refer to Appendix E wherewediscuss formal aspectsofthe BNF (mainly in the (AA)-case) and give a proof of the lemma in Section E.3. 6.3. Approximate BNF. 6.3.1. Elliptic ﬁxed point case ((CC)-Case). — Our aim is to give a more quantitative version of Proposition 6.1. h 1/2 2 Recall that W ={(z,w) ∈ D(0, e ρ ) , −izw ∈ D(0,ρ)} and we denote h,D(0,ρ) sometimes by W the set W . h,ρ h,D(0,ρ) Let m ≥ 4 be an integer. Applying Proposition 6.1 with m = N − 1 we can assume that the diffeomorphism f is of the form f = ◦ f 0 0 (6.140) 2 2m (r) = 2πω r + O (r), and F (z,w) = O (z,w). 0 0 0 In particular (cf. Remark 2.1)for some h > 0and any ρ> 0 small enough we can assume that (6.141) F h ρ . e W h,D(0,ρ) It states that if a polynomial of degree n is bounded above by some constant M on a not pluripolar compact set m m K ⊂ C then its size at any point z ∈ C is not larger than M × exp(ng (z)) where g (z) is the Green function of K with K K pole at ∞. 58 RAPHAËL KRIKORIAN Denote by (p /q ) the sequence of best rational approximations of ω which has n n n≥1 0 the following properties (cf. [20], Chap. 5, formulae (7.3.1)–(7.3.2) and Prop. 7.4): for all n ∈ N 1 1 (6.142) <(−1) (q ω − p )< , n 0 n q + q q n n+1 n+1 and (6.143) ∀ 0 < k < q , ∀ l ∈ Z, |kω − l|≥|q α − p | > . n 0 n−1 n−1 2q We refer to Notations 2.3, 2.6 and 4.8 before stating the following proposition. Proposition 6.4. — Let a := max(2a + 1, 30) where a is the exponent that appears in Lemma 5.4 and assume that (6.141) holds for some m ≥ a . Then for any β> 0 and any n 1 1 β 1−β BNF BNF q BNF there exist g ∈ Symp (W −6 ), and functions F ∈ O (W −6 ) ∩ O (z,w), ∈ −1 −1 σ −1 ex.,σ h,q h,q n n q q q n n n −6 O (D(0, q )) such that BNF −1 BNF −6 BNF BNF (6.144) [W ] (g ) ◦ ◦ f ◦ g = ◦ f −1 F −1 h,q 0 0 F q q n n −1 −1 q q n n 1−β BNF q (6.145) (r) − BNF(f )(r) = O (r), in R[[r]] −1 BNF (6.146) 1 −1 BNF −(m−27) (6.147) g − id 1 ≤ q −1 C BNF 1−β (6.148) F ≤ exp(−q ). −1 W −6 n h,q BNF If ∈ TC(A, B) (see Notation 2.6) one can choose ∈ TC(2A, 2B). −1 Proof. — See the Appendix, Section F.2. 6.3.2. (AA) or (CC) case when ω is Diophantine. — We formulate here a more quan- titative version of the classical Birkhoff Normal Form Theorem (Propositions 6.1, 6.2) which holds both in the (AA) or (CC) cases, provided ω is Diophantine: (6.149) ∀ k ∈ Z {0}, min|kω − l|≥ (τ ≥ 1). l∈Z |k| CC AA Let as usual W be equal to either W or W and ∈ O (D(0, 1)), h,D(0,ρ) σ h,D(0,ρ) h,D(0,ρ) (r) = 2πω r + O(r ),where ω is assumed to be Diophantine with exponent τ . 0 0 We deﬁne (as before a is the constant introduced in Lemma 5.4) (6.150) a := max(2(τ + a), 12) 1,τ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 59 andweassume that forsome m ≥ a , the function F ∈ O (e W ) (h > 0) satisﬁes 1,τ σ h,D(0,1/2) 2m (CC) − Case : F(z,w) = O (z,w) (6.151) (AA) − Case : F(θ , r) = O(r ). We set (CC) − Case : b = 2(τ + 1) (6.152) (AA) − Case : b = τ + 1. Proposition 6.5. — Assume that ω satisﬁes (6.149) and that for some m ≥ a (6.151) 0 1,τ BNF b BNF holds. Then, for any β> 0 and any 0 <ρ 1, there exist ∈ O (D(0,ρ )), F ∈ β σ ρ ρ 1−β (1/ρ) BNF O (W b ) ∩ O (r) and g ∈ Symp (W b ) such that on W b one τ τ τ σ h,D(0,ρ ) h,D(0,ρ ) h,D(0,ρ ) ρ ex.,σ has BNF −1 BNF BNF BNF (6.153) (g ) ◦ ◦ f ◦ g = ◦ f ρ ρ ρ ρ 1−β BNF (1/ρ) (6.154) (r) − BNF(f )(r) = O (r), in R[[r]] BNF BNF m−10 g − id 1 ≤ ρ BNF 1−β (6.155) F exp(−(1/ρ) ). ρ b h,D(0,ρ ) BNF If ∈ TC(A, B) then ∈ TC(2A, 2B). Proof. — See the Appendix, Section F.3. Remark 6.6. — Inequality (6.155) can also be written BNF (1−β)/b F exp(−(1/ρ) ). 1/β W τ h,D(0,ρ) We note that Iooss and Lombardi (Theorem 1.4 of [24]) obtained, for a similar problem, a more precise estimate but with essentially the same exponent 1/b = 1/(1 + τ) (their 2 1/(1+τ) estimate reads ≤ (cst)ρ exp(−(cst)/ρ )). Remark 6.7. — In the (CC)-case and when ω is in DC(κ, τ ),one canprove the previous proposition (maybe not with the same value for the exponent b ) by using Propo- −1 τ sition 6.4 and the fact that q ≤ q ≤ κ q . n n+1 6.4. Consequence of the convergence of the BNF. Lemma 6.6. — Assume that BNF(f ) coincides as a formal power series with a holomorphic function ∈ O(D(0, ρ)) and, for 0 <ρ ≤ ρ,let ∈ O(D(0,ρ)) be such that N+1 (r) − BNF(f )(r) = O (r) in R[[r]] (6.156) ≤ 1. D(0,ρ) 60 RAPHAËL KRIKORIAN Then −1 − exp(−N). D(0,e ρ) ∞ ∞ N k k k Proof. — Let (z) = ξ z , (z) = b z , = ξ z and = k k N k N k=0 k=0 k=0 b z .Wehavefrom(6.156) and the fact that = BNF(f ) in R[[r]] k=0 (6.157) = . N N On the other hand, we observe that if g : z → g z is in O(D(0,ρ)) one has by k∈N k −1 Cauchy’s estimates |g |ρ ≤g ,hence for |z| < e ρ k D(0,ρ) k k g z ≤ g (z/ρ) k D(0,ρ) k≥N+1 k≥N+1 −N ≤ 2e g . D(0,ρ) As a consequence, −N −N −1 −1 − e , − e . N D(0,e ρ) D(0,ρ) N D(0,e ρ) D(0,ρ) We conclude using (6.157). To summarize, Corollary 6.7. — If BNF( ◦ f ) converges and coincide on D(0, ρ) with ∈ O(D(0, ρ)), then for any β> 0 and ρ 1 one has: –If ω is τ -Diophantine ((AA) or (CC)-case) BNF 1−β − b exp(−(1/ρ) ). D(0,ρ ) – In the (CC) case for any ω irrational 1−β BNF − −6 exp(−q ). −1 D(0,q ) n+1 n+1 n+1 7. KAM Normal Forms We present now, in the uniﬁed (AA)-(CC) framework, the KAM scheme that is central in all this paper. This will be used in Sections 10 and 11 to construct the adapted Normal Forms and in Section 12 to get estimates on the Lebesgue measure of the set of KAM circles. For the sake of clarity we break down our main result into three proposi- tions: Propositions 7.1, 7.2, 7.4. As usual we denote in the (AA)-case M = T × C,M = T× R,O = T×{0} and ∞ R in the (CC)-case M = C × C and M = M∩{r ∈ R},O={(0, 0)}. R ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 61 7.1. The KAM statement. — Let 0 < ρ< h/2 < 1/2, A ≥ 1, B ≥ 1and ∈ O (e D(0, ρ)) satisfying the following twist condition (see (2.59)): −1 2 3 (7.158) ∀ r ∈ R, A ≤ (1/2π)∂ (r) ≤ A, and (1/2π)D ≤ B. −1 h Let ω(r) = (2π) ∂(r). The image of D(0, e ρ) by ω is contained in a disk D(ω (0), 3Aρ). We can assume without loss of generality that ω(0) ∈[−1/2, 1/2] and consequently, if ρ is small enough we can assume (7.159) ω(D(0, e ρ) ∩ R)⊂[−3/4, 3/4]. Let C, a be the constants of Proposition 5.5. We introduce (7.160) a = 2(a + 2) + 10 2 0 and assume that F ∈ O (e W ) satisﬁes σ h,D(0,ρ) (7.161) F h ≤ ρ . e W h,D(0,ρ) By Cauchy’s inequality (2.53) one has j 2(a +2)+1 (7.162) ε := maxD F ≤ ρ . h,D(0,ρ) 0≤j≤3 Associated to this ε> 0 there exists a unique N > 0such that − ln ε = N/(ln N) . We then deﬁne for n ≥ 1 the following sequences that depend on ε = ε , h and ρ> 0: n−1 N = (4/3) N −hN /(ln N ) n n ε = e ⎪ n 2(a +2) −1 0 (7.163) K = ε ,(a ≥ 5) ⎪ −2 δ = 2(ln N ) h n n n−1 n−1 ρ = ρ exp(− δ ), h = h − (1/2) δ > h/2. n j n j j=1 j=1 If ρ is small enough, for all n ≥ 1 one has −1/20 −1/20 ρ ≥ e ρ, h ≥ e h n n and (cf. (7.162)), −1 (7.164) ρ /2 > 2K −1 −1 −a (7.165) (δ (2K )) K ε < C n n n (C is the constant of Proposition 5.5). 62 RAPHAËL KRIKORIAN Proposition 7.1. — Assume that and F are as above and that ρ 1. Then, with the A,B notations (7.163) the following holds: for n ≥ 1 there exist a decreasing (for the inclusion) sequence of holed domains (U ) , functions ∈ O (U ), F ∈ O (W ) with U = D(0, ρ), = , n n≥1 n σ n n σ h ,U 1 1 n n F = F and, for n ≥ 2, 1 ≤ m < n, diffeomorphisms g ∈ Symp (W ), such that: 1 m,n h ,U n n ex.,σ (7.166) satisﬁes a (2A, 2B) − twist condition, (cf. (2.59)) (7.167) g (W ) ⊂ W m,n h ,U h ,U n n m m −1 (7.168) on W , g ◦ ◦ f ◦ g = ◦ f h ,U F m,n F n n m,n m m n n 1/2 (7.169) g − id 1 ≤ ε , m,n C (7.170) maxD F ≤ ε . n W n h ,U n n 0≤j≤3 Proof. — We construct inductively for n ≥ 2 sequences U , F , , g satisfying the n n n m,n conclusion of the proposition with the additional requirements Requirement 1: For n ≥ 2, U is of the form (7.171) U = D(0,ρ ) D(c ,κ ), c ∈ R, #I ≤ 2N n n i i i n n−1 i∈I n−1 n−1 n−1 −1 −1 δ 2 1/2 δ −1 l l l=1 l=1 (7.172) K ≤ κ ≤ K e ,( κ ) ≤ 2e N K . i l n−1 1 i l i∈I l=1 Requirement 2: For n ≥ 2, ∈ O (U ) satisﬁes an (A , B )-twist condition with (see n σ n n n (2.52) for the notation a(U )) −1 −1 (7.173) 1 ≤ A ≤ 2A − K , 1 ≤ B ≤ 2B − K n n n n (7.174) 8max(ρ, a(U )) × A × B < 1. n n n n−1 1/2 1/2 (7.175) − 3 ≤ ε ≤ 2ε C (D(0,ρ)) n l 1 l=1 n−1 1/2 (7.176) and ∀ m < n, g − id 1 ≤ C ε ε (Cfrom (2.43)). m,n C l l=m For some n ≥ 1, assume the existence of U , F , and the validity of conditions n n n (7.171), (7.172), (7.173), (7.174), (7.175)(if n ≥ 2) and deﬁne ω = (1/2π) . Since (7.174) n n −1 is satisﬁed we can apply Proposition 2.7 (with A = A ,B = B ,3A ν = K , β = l/k): for n n n 2 −1 each (k, l) ∈ Z ,0 < k < N ,suchthat D(l/k,(3A K ) ) ∩ ω (U ) = ∅, there exists n n n n n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 63 (n) c ∈ R such that l/k (n) ω (c ) = l/k l/k (7.177) (n) −1 −1 ∀ r ∈ C D(c , K ), |ω (r) − (l/k)|≥ (3A K ) . n n n l/k n We denote E ={(k, l) ∈ Z , 0 < k < N , 0≤|l|≤ N , n n n −1 D(l/k,(3A K ) ) ∩ ω (U ) = ∅} n n n n and we see that (7.178)#E ≤ 2N . Note that from (7.175)and (7.159)wehave |l/k|≤ 1. Hence, if we deﬁne (n) −1 (7.179)V = U D(c , K ) n n l/k (k,l)∈E we have for any r ∈ V (cf. (7.173)) ∗ −1 ∀ (k, l) ∈ N × Z, 1 ≤ k < N %⇒ k ∂ (r) − l ≥ (6AK ) ; n n n 2π hence the non-resonance condition (5.126) (with τ = 0, K = 6AK ,N = N )issatisﬁed. n n On the other hand (7.171)–(7.172)(n ≥ 2) and (7.164)(n = 1) show using (7.179)that (recall ρ < ρ< h/2) −1 −1 (7.180)d(W ) = d(V ) = min(d(U ), K ) = K h ,V n n n n n n and (7.165)and (7.170) show that −1 −a (7.181) (δ d(W )) K F < C . n h ,V n n h ,U n n n n We can thus apply Proposition 5.5 (with τ = 0, K = 6AK , δ = δ ,N = N )on V :ifone n n n n deﬁnes (n) −δ −δ δ −1 n n n (7.182)U = e V = e U D(c , e K ) n+1 n n l/k n (k,l)∈E −δ /2 n there exist Y ∈ O(e W ),F ∈ O (W ), ∈ O (U ) such that n n+1 σ h ,U n+1 σ n+1 h ,V n+1 n+1 n n (ρ small enough) −1 (7.183) Y −δ /2 K δ F n e W n n W h ,V n h ,U n n n n −1 (7.184) W , f ◦ ◦ f ◦ f = ◦ f h ,U Y F F n+1 n+1 n n n Y n+1 n+1 n 64 RAPHAËL KRIKORIAN (7.185) = + M(F ) n+1 n n j −1 − a 2 −δ N /2 0 n n (7.186) maxD F K (δ K ) (F + e F ). n+1 W A n n n n W h ,U n W h ,U n n n+1 n+1 h ,U n n 0≤j≤3 Let us show that the Requirements 1 (7.171)–(7.172) are satisﬁed for n + 1. From (7.182)and (7.171) we see that U = D(0,ρ ) D(c ,κ ) n+1 n+1 i i i∈I n+1 −1 2 δ δ −1 n n where I ≤ I + 2N (cf. (7.178)) and for all i ∈ I ,min(e K , e K ) ≤ κ ≤ n+1 n n+1 i n n−1 n −1 2 2 δ 1/2 δ 1/2 2 −2 1/2 l n l=1 K e . Similarly, ( κ ) ≤ e (( κ ) + (2N K ) ).Inother words, 1 i∈I i i∈I i n n n+1 (7.171)–(7.172) are satisﬁed for n + 1. Let us now prove that the Requirements 2, (7.173), (7.174)(7.175) are satisﬁed for Wh n + 1 and in particular that has a nice Whitney extension := .Weﬁrst n+1 n+1 n+1 3 Wh apply Lemma 2.2 to get a C , σ -symmetric extension M(F ) : C → C for (M(F ), U ) n n n such that j Wh 3 −6 j sup D M(F ) (1 + #J ) (δ d(U )) maxD M(F ) −δ /10 . n C U n n n e U n n 0≤j≤3 0≤j≤3 In particular, using Cauchy’s inequalities, (7.171), (7.172), (7.163), (5.104)one gets j Wh 6 −1 −6 −3 (7.187) sup D M(F ) N (δ K ) δ M(F ) n C n n U n−1 n−1 n n 0≤j≤3 7 1/2 K ε ≤ ε . n−1 n From (7.185) we see that if we deﬁne the σ -symmetric function Wh (7.188) := + M(F ) n+1 n n one has n+1 n+1 n+1 −1 −1 1/3 and (7.173) , are satisﬁed (since −K + ε < −K ). To see that (7.174) holds we n+1 n+1 n n n+1 use the fact that since the second inequality in (7.172) is true for n+ 1 (as already checked) n −1/2 2 1/2 −1 one has a(U ) ≤ ( κ ) ≤ 2 N K ≤ K .If ρ is small enough we see n+1 l i l 1 i∈I l=1 n+1 that (7.162), (7.163)and (7.173) ensure the validity of (7.174) . n+1 n+1 −(3/4)δ Finally let us check (7.176) . From Lemma 2.2 we see that (Y , e W ) n+1 h ,V n n 3 Wh has a C σ -symmetric Whitney extension Y such that Wh 3 −6 j 3 −(2/3)δ (7.189) Y (1 + #J ) (δ d(U )) maxD Y n . V n n n C e V n n n 0≤j≤2 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 65 2 −1 From (7.171), (7.179), (7.172) we see that #J ≤ 2N ,d(V ) ≥ K hence using Cauchy’s V n n n n (1/2)+ −1 0+ 7 inequalities, (7.183), (7.163) and the fact that δ , N ≤ K and K ε ≤ ε ,weget n n n n n n Wh 1/2 (7.190) Y 3 ≤ ε . n n −1 If we deﬁne g = f ∈ Symp (W ) and for m ≤ n, g = g ◦ g n,n+1 Wh h ,U m,n+1 m,n n,n+1 ex,σ n+1 n+1 we have from (4.98)and (2.43) g − id 1 ≤ C(g − id 1 +g − id 1 ) ≤ C ε ε m,n+1 C m,n C n,n+1 C l m l=m which is (7.176) and implies (7.169) . n+1 n+1 −1 −1 Note that f = f on W and (7.184)shows that (7.168) and (7.167) Wh h ,U Y n+1 n+1 n+1 n+1 Y n are satisﬁed. We now check that (7.170) holds for n + 1; from (7.186) it is enough to verify a +2 2 −δ N /2 0 n n (7.191)K (ε + e ε )< ε n n+1 n n −δ N /2 n n 2(a +2) or equivalently since e = ε ,K = ε , n n n 2−(1/2) 2ε ≤ ε n+1 which is clearly satisﬁed since 3/2 > 4/3, cf. (7.163). 7.2. Localization of the holes. — We can localize the holes of the domains U : Proposition 7.2 (Localization of the holes).— For each 1 ≤ m < n, one has 1/2 (7.192) ∂ − ∂ 2 ε n m m and for some sets E ⊂{(k, l) ∈ Z , 0 < k < N , 0≤|l|≤ N } (1 ≤ i ≤ n − 1) one can write U i i i n as n−1 n−1 (i) −1 δ t=i (7.193) D(0,ρ ) D(c , s K ), s = e ∈[1, 2] n i,n−1 i,n−1 l/k i i=1 (k,l)∈E (i) (i) −1/5 where ρ ≥ e ρ and c is on the real axis and is the unique solution of the equation ω (c ) := l/k i l/k (i) −1 (2π) ∂ (c ) = l/k. l/k Proof. — Inequality (7.192) is consequence of (7.188), (7.187). The expression (7.193) comes from (7.182). We now give a more detailed description of the structure of D(U ), the set of holes of the domains U appearing in Proposition 7.2, cf. (7.193). n 66 RAPHAËL KRIKORIAN Lemma 7.3. — With the notations of Propositions 7.1–7.2: (1) For any n ≤ n , (k , l ) ∈ E ,j = 1, 2, one has 1 2 j j n (n ) (n ) 1 2 1/2 if l /k = l /k then |c − c | ε 1 1 2 2 l /k l /k n 1 1 2 2 1 (7.194) (n ) (n ) 1 2 −2 if l /k = l /k then |c − c | N . 1 1 2 2 l /k l /k n 1 1 2 2 (2) Let n , n ∈ N,n ≤ n and 0 <κ <κ be such that 1 2 1 2 2 1 −2 1/2 κ + κ N , ε κ − κ . 1 2 1 2 n n 2 1 (n ) Then, two disks D(c ,κ ), (k , l ) ∈ E ,j = 1, 2, are either disjoint or l /k = l /k and j j j n 1 1 2 2 l /k j j j (n ) (n ) 2 1 D(c ,κ ) ⊂ D(c ,κ ). 2 1 l /k l /k 2 2 1 1 Proof. — Item (1) is due to (7.192) and the fact that if l /k = l /k 1 1 2 2 −2 |(l /k ) − (l /k )|≥ 1/(k k ) ≥ N . 1 1 2 2 1 2 −2 Item (2) is a consequence of Item (1). Indeed, if l /k = l /k then since κ + κ N and 1 1 2 2 1 2 (n ) (n ) (n ) (n ) 1 2 −2 1 2 |c − c | N (we assume n ≤ n ), the disks D(c ,κ ) and D(c ,κ ) must have 1 2 1 2 l /k l /k n l /k l /k 1 1 2 2 2 1 1 2 2 an empty intersection. On the other hand, if l /k = l /k , then because of the fact that 1 1 2 2 (n ) (n ) (n ) (n ) 2 1/2 2 1/2 1 1 |c − c | ε ,the disk D(c ,κ ) contains D(c ,κ ) because ε + κ <κ . 1 2 2 1 l /k l /k n l /k l /k n 1 1 2 2 1 1 1 2 2 1 7.3. Whitney conjugation to an integrable model. — By applying Lemma 2.2 one sees −δ −δ 3 n n that (F , e W ) and (f , e W ) have C real symmetric Whitney extensions n h ,U F h ,U n n n n n Wh −δ Wh F ∈ O (e W ), f Wh ∈ Symp (W ) (the canonical map associated to F )such σ h ,U F h ,U n n n ex,σ n n n that (see the discussion leading to (7.187) and inequality (4.98)) Wh 1/2 1/3 3 Wh 1 F ε , f − id ε . C F C n n n We hence have −δ −1 (7.195)on e W , g ◦ ◦ f Wh ◦ g = ◦ f Wh . h ,U F m,n F n n m,n m m n n We show in the next Proposition that shrinking a little bit the domain of validity of the preceding formula one can impose that g leaves invariant the origin O={r = 0}∩ M . m,n R Lemma 7.4. — There exists g ∈ Symp (W −1 ) that coincides with g on m,n m,n ex,σ h /2,U D(0,K ) n n m W −1 and h /2,CD(0,K ) n m 1/4 (7.196) g ({r = 0})={r = 0}, g − id 1 ≤ ε . m,n m,n C −1 −1 Wh 3 −(1/2)δ Proof. — Recall that g = f ◦ ··· ◦ f with Y ∈ C ∩ O (e W ) m,n Wh Wh σ h ,V k k Y k n−1 satisfying (7.190). Let χ : R →[0, 1] be a smooth function with support in [−1, 1] ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 67 and equal to 1 on [−1/2, 1/2] and deﬁne the C σ -symmetric function Y = (1 − 2 Wh Wh 3 Wh 1/4 χ((K r/2) ))Y . One has Y = Y on W and Y 3 K Y 2 ≤ ε m k h,CD(0,K /2) k C C k k m m 1/4 −1 −1 −1 −1 1 hence f coincide with f on W and f − id ε . Since for m ≤ k ≤ Wh C h,CD(0,K ) k Y Y Y k k −1 n − 1, Y is null on a neighborhood of {r = 0} the diffeomorphism f ﬁxes {r = 0} and n−1 1/4 1/4 so does g . The inequality in (7.196) follows from the fact that ε ε . m;n k=m m Note that the sequence of diffeomorphisms n → g converges in C to a σ - m,n symmetric diffeomorphism g : C → C ﬁxing the origin and that satisﬁes g − m,∞ m,∞ id 1 ε . On the other hand, the sequence of diffeomorphisms (f Wh ) converges in C C m F n to the identity and from (7.192) the sequence of functions ( ) , ∈ O (U ) converges n n σ n 2 2 in C to some σ -symmetric limit ∈ C (C);hence from (7.195) ∞ σ −δ −1 on e W −1 , g ◦ ◦ f Wh ◦ g = . F m,∞ h /2,U D(0,K ) m,∞ n n m m m ∞ n≥m Recall the notations of Section 4.4 and let −δ −1 L = R ∩ e (U D(0, K )), m n n≥m −δ −1 W = M ∩ e W . L R m h /2,U D(0,K ) n n m n≥m Proposition 7.5. — For any m ≥ 1 one has −1 (7.197) on W , g ◦ ◦ f ◦ g = L F m,∞ m,∞ m m ∞ (7.198) g (W ) ⊂ W , m,∞ L R∩U 1/4 (7.199) g ({r = 0})={r = 0}, g − id 1 ≤ ε m,n m,n C 2(a +3) −2δ m 0 (7.200) Leb ((R ∩ e U ) L ) ε . R m m Proof. — Let us prove (7.198). Note that since g and g coincide on m,n m,n −δ W −1 one has from (7.167) g (e W −1 ) ⊂ W hence since g is m,n h ,U m,n h /2,CD(0,K ) h ,U D(0,K ) m m n m n n m σ -symmetric, g (W ) ⊂ W and g (L ) ⊂ W = W . m,n L R∩U m,∞ m R∩U m m m R∩U The conjugation relation (7.197) comes from the fact that Wh ◦ f Wh coincides on m m W with ◦ f . R∩U m m m For the proof of (7.200) we ﬁrst observe that from the expression (7.193), for each − δ −δ 2 −1 l n l=m n > m the set e U e U is a union of at most 2N disks of radii ≤ 2K m n n n 2 −1 hence the Lebesgue measure of its intersection with M is ≤ 4N K . In consequence, n n ∞ 2(a +3) − δ −δ 2 −1 0 l n l=m the Lebesgue measure of R ∩ e U e U is N K ≤ ε m n n≥m+1 n=m+1 n n 68 RAPHAËL KRIKORIAN hence 2(a +3) −2δ −δ 0 m n Leb (e U e U ) ε M m n n≥m −δ δ −1 −2δ n m m and since L ⊃ ( e U ) e D(0, K ) we get that Leb ((R ∩ e U ) L )) ≤ n M m m n≥m m 2(a +3) 0 δ −1 εm + e K ;(7.200) follows from this inequality. Remark 7.1. — If U is a holed domain, Propositions 7.1, 7.2, 7.5 as well as their proofs, extend without any change to the situation where F ∈ O (e W ) and σ h,U ∈ O (e U) satisﬁes the twist condition (7.158)–(2.60) and if the following smallness assumption on F holds (7.201) F h ≤ d(W ) . e W h,U h,U 8. Hamilton-Jacobi Normal Form and the Extension Property Our aim in this section is to provide a useful approximate Normal Form (that we call the Hamilton-Jacobi Normal Form) in a neighborhood of a q-resonant circle {r = c} (by which we mean that for some (p, q) ∈ Z × N , p ∧ q = 1 ω(c) = ). Let 0 < ρ< h/2 < 1/20, c ∈ R, (p, q) ∈ Z × N , p ∧ q = 1, ∈ O (D(c, 6ρ) ), F ∈ O (W ) such that σ h,D(c,6ρ) −1 −1 2 −1 3 (8.202) ∀ r ∈ R, A ≤ (2π) ∂ (r) ≤ A, and (2π) D ≤ B. a −8 (8.203) ε := F ≤ min((6ρ) ,(10A) ), h,D(c,6ρ) −1 (8.204) ω(c) := (2π) ∂(c) = 1/8 −1 (8.205) (6ρ) <(Aq) < h/10, 6ρ< |c|/4 where a is the constant appearing in Proposition G.1 of Appendix G on Resonant Nor- mal Forms. The purpose of this section is to prove the following result: Proposition 8.1 (Hamilton-Jacobi Normal Form).— Let D = D(c, ρ) . There exists a disk D, 1/33 (8.206) D := D(q c, ρ) q ⊂ D = D(c, ρ) , with ρ q≤ ε and HJ HJ HJ ∈ O (D D)), F ∈ O (W ), g ∈ Symp ((W ) σ σ q q h/9,(DD) σ h/9,(DD) D D D ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 69 such that HJ (8.207) satisﬁes a (2A, 2B) − twist condition HJ HJ −1 (8.208) W ,(g ) ◦ ◦ f ◦ g = HJ ◦ f HJ q F h/9,(DD) F D D D D HJ 1/8 (8.209) g − id 1 qε HJ 1/4 (8.210) F exp(−1/(6qρ) )ε. D h/9,(DD) Moreover, one has the following: HJ Extension property: ( , D, D) satisﬁes the following Extension Principle: If there exists a holomorphic function ∈ O(D) such that HJ − ν (4/5)D(1/5)D 1/200 then ρ q ν . Remark 8.1. — From Lemma K.1 and Remark K.1 of the Appendix, we just have to prove the Proposition in the (AA)-setting. This is the setting in which we shall work in all this section. The proof of the ﬁrst part of Proposition 8.1 is done in Section 8.7 and that of the second part (Extension Principle), based on Proposition 8.10, is done in Section 8.9. From now on we deﬁne ρ = 6ρ. 8.1. Putting the system into Resonant Normal Form. — From Proposition G.1 of the Appendix on the existence of approximate q-Resonant Normal Form, we know that there res −1/q −1/q cor −1/q exist ∈ O (D(c, e ρ)), g ∈ Symp (e W ), F , F ∈ O (e W ) σ RNF h,D(c,ρ) σ h,D(c,ρ) ex,σ res res such that F is 2π/q-periodic, M (F ) = 0, and −1 −1/q res cor e W , g ◦ ◦ f ◦ g = ◦ ◦ f ◦ f h,D(c,ρ) F RNF 2π(p/q)r F F RNF (8.211) res res F is 2π/q − periodic, M (F ) = 0, with − ( − 2π(p/q)r) −1/q F D(c,e ρ) W h,D(c,ρ) res −1/q F F e W h,D(c,ρ) h,D(c,ρ) (8.212) −1/4 cor F −1/q exp(−ρ )F e W W ⎪ h,D(c,ρ) h,D(c,ρ) −1 g − id 1 ≤ (qρ ) F RNF C h,D(c,ρ) 70 RAPHAËL KRIKORIAN Inequalities (8.212) and the fact that satisﬁes an (A, B)-twist condition on D(0, ρ) show that there exists a unique c ∈ R such that ∂ (c) = 0, |c − c| ε. 8.2. Coverings. — We denote R = R + −1[−h, h] and by j the q-covering h q j (C/(2π Z)) × C → (C/(2π/q)Z) × C (8.213) (θ + 2π Z, r) → (θ + (2π/q)Z, r). res −2/q Since the function F : (θ , r) : (R /(2π)Z) × D(c, e ρ) → C is invariant by h−2/q (θ , r) → (θ + 2π/q, r) onecan pushitdowntoafunction res res res −2/q F : (R /(2π/q)Z) × D(c, e ρ) → C, F ◦ j = F . h−2/q q j j q q Let : (C/(2π/q)Z) × C → C/(2π)Z × C (8.214) (θ , r) → (qθ, q(r − c)) res −2/q and deﬁne F : (R /(2π)Z) × D(0, e qρ) → C by qh−2 res res 2 −1 F = q F ◦ ; j q −2/q −2/q for all (θ, r) ∈ T × D(0, qe ρ) and (θ , r) ∈ T × D(c, e ρ) such that θ = qh−2 h−2/q qθ, r = q(r − c) one has res res 2 (8.215) F (θ, r) = q F (θ , c + r). Let f res be the (exact) symplectic mapping (for the symplectic form dθ ∧ d r)deﬁnedby (4.87): if ( ϕ, R) = (ϕ, R), (θ, r) = (θ , r) res ( ϕ, R) = f res (θ, r) ⇐⇒ (ϕ, R) = f (θ , r). F F If we set (8.216) (r) := q (c + (r/q)) − 2π(p/q)(r/q) 2 2 3 = (1/2)∂ (c)r + O(r ) 2 3 = r + r b(r) we have −1 res (8.217) ◦ ◦ f ◦ = ◦ f res . q F F q q ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 71 Note that since satisﬁes an (A, B)-twist condition, one has from the ﬁrst equation of (8.212) (to which one applies Cauchy’s inequality), the estimate −1/10 2 (8.218) ∀ r ∈ D(0, e ρ), ∂ (r) 1, 3 −1/10 1. C (D(0,e qρ)) 8.3. Approximation by a Hamiltonian ﬂow. — The following proposition says that up to some very good approximation ◦ fres can be seen as the time-1 map of a Hamilto- nian vector ﬁeld in the plane. vf per −2/q Proposition 8.2. — There exists F , F ∈ O (T −2/q × D(0, e qρ/2)),suchthat σ e qh/2 −2/q on T −2/q × D(0, e qρ/2) one has e qh/2 (8.219) ◦ fres = per ◦ fvf F +F F per res 1/4 res 2 (8.220) F = F + O(ρ F −2/q ) = O(q F ) T ×D(0,e qρ) h,D(0,ρ) −2/q e qh vf 1/4 −2/q −2/q (8.221) F exp(−1/(qρ) )F . h,D(0,ρ) e qh/2,D(0,e qρ/2) Proof. — This is a consequence of (8.212), (8.215) and Proposition H.1 applied to ◦ f res (since by (8.212), (8.215), condition (H.545) is satisﬁed). i 3 Let F(θ , r) = f (θ )r + r f (θ , r) and deﬁne (cf. (8.216)) i=0 per (8.222) (θ , r) = (r) + F (θ , r) 2 2 3 (8.223) =: r + f (θ ) + f (θ )r + f (θ )r + r (b(r) + f (θ , r)) 0 1 2 1 f (θ ) 1 f (θ ) 1 1 = ( + f (θ )) r + − 2 + f (θ ) 4 + f (θ ) (8.224) 2 2 + f (θ ) + r (b(r) + f (θ , r)) where −3 2 (8.225)max (|f |,|f |,|f |,|f |) (qρ) q ε. 0 1 2 −2/q T ×D(0,e qρ/2) −2/q e qh/2 8.4. From to .— We assume in the rest of this section that > 0 and we set (8.226) ρ = qρ/3. The next lemma provides a more convenient expression for the function, viewed per as a Hamiltonian, = + F which was deﬁned in (8.222). 72 RAPHAËL KRIKORIAN Lemma 8.3. — There exists a (not exact) symplectic change of coordinates G∈Symp (T × qh/3 −1/10 D(0,ρ )) of the form G(θ , r) = (θ , r − e (θ )) and ∈ O(T × D(0, e ρ )) such that q 0 qh/3 q −1 2 3 (8.227) (θ , r) := ◦ G (θ , r) = (θ)(r − e (θ ) + r f (θ , r)) with , e , e ∈ O (T ),f ∈ O (T × D(0,ρ )), 0 1 σ qh/3 σ qh/3 q −2 −1 (8.228) (·)− qρ ε, max(e ,e ), qρ ε, f 1. qh/3 0 qh/3 1 qh/3 qh/3,ρ Proof. — See Appendix L.1. Remark 8.2. — The previous lemma and (8.224) show that (θ) = + f (θ ) + O(ρ ε) and 1 f (θ ) e (θ )=− + O(ρ ε), 2 + f (θ ) 1 f (θ ) f (θ ) 1 0 e (θ )=− + + O(ρ ε). 4 ( + f (θ )) + f (θ ) 2 2 Remark 8.3. — Since is deﬁned up to an additive constant (this will not change the value of e ), we can assume that (θ , e (θ )) dθ = 0 1/2 (θ) 2π which is equivalent to the following condition that we will assume to hold from now on dθ 1/2 (8.229) (θ) e (θ ) = 0. 2π 8.5. Hamilton-Jacobi Normal Form for .— The symplectic diffeomorphism is the time-1 map of a Hamiltonian deﬁned on the cylinder, and as such, it is integrable in the Hamilton-Jacobi sense: the level lines of the Hamiltonian foliate the cylinder and naturally provide invariant curves for the Hamiltonian ﬂow. On some open sets it is possible to conjugate to a Hamiltonian depending only on the action variable: this is the Hamilton-Jacobi Normal Form; see Proposition 8.7. The purpose of this Subsection is to quantify this fact. These are cylindrical domains outside the “eyes” deﬁned by separatrices (think of a pendulum). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 73 Recall the expression for 2 3 (θ , r) = (θ)(r − e (θ ) + r f (θ , r)). Let 0 ≤ s ≤ h/3. We denote −1 (8.230) ε := e 0 qρ ε, ε = ε (s)=e , 1 1 C (T) 1,s 1 1 qsh/3 and for L 1 we introduce 1/2 1/2 (8.231) λ := Lε ,λ = λ(s, L) = Lε , 0,L s,L 1 1,s with the requirement −1/2 (8.232) λ < qρ/6 = ρ /2 or equivalently 1 L qρε . s,L q 1,s We notice that 0 <λ ρ and that from the Three Circles Theorem s,L q 1−s s (8.233) ε (0) ≤ ε (s) ≤ ε (0) ε (1) 1 1 1 1 hence 1/2 (1−s)/2 (8.234) λ = Lε ≤ λ ≤ Lε . 0,L s,L 1 1 Notation 8.4. — For 0 < a < a and z ∈ C we denote by A(z; a , a ) the annulus centered 1 2 1 2 at z with inner and outer radii of sizes respectively a and a .When z = 0 we simply denote this annulus 1 2 by A(a , a ). 1 2 Before giving the Hamilton-Jacobi Normal Form of we need two lemmas. Lemma 8.5. — There exists a holomorphic function g deﬁned on Dom(g) := (T × A(λ ,ρ )) qsh/3 s,L q 0≤s≤1 such that for every (θ , z) ∈ Dom(g) one has (8.235) (θ , g(θ , z)) = z . Moreover, there exists g ˚ ∈ O(Dom(g)) such that on Dom(g) one has −1/2 −2 ˚ ˚ (8.236) g(θ , z) = (θ) z(1 + g(θ , z)), g L . Dom(g) Proof. — See the Appendix, Section L.3. 74 RAPHAËL KRIKORIAN Since T × A(λ ,ρ ) ⊂ Dom(g) we can deﬁne the function ∈ O(A(λ ,ρ )) by 0,L q 0,L q : A(λ ,ρ ) → C 0,L q 2π −1 (8.237) (u) = (2π) g(ϕ, u)dϕ. Using (8.236) we see that can be written 2π −1 −1/2 (u) = γ u(1 + (u)), γ := (2π) (θ) dθ, −2 L . A(λ ,ρ ) s,L q Lemma 8.6. — There exists a solution H ∈ O(A(2λ ,ρ /2)) of the equation s,L q (8.238) (H(z)) = z. Moreover it can be written −1 −2 ˚ ˚ (8.239)H(z) = γ z(1 + H(z)), H ≤ L . A(2λ ,(1/2)ρ ) s,L q Proof. — See the Appendix, Section L.4. We now apply the preceding results with s = 1/2. Proposition 8.7 (Hamilton-Jacobi).— There exists an exact symplectic change of coordinates 1/32 W ∈ Symp (T × A(3ε ,ρ /3)) such that qh/7 q ex,σ 1 −1 (8.240) W ◦ ◦ W = 2 1/4 (8.241) W − id 1 qε . Proof. — Let H be the function deﬁned by the previous lemma (with s = 1/2) and deﬁne for z ∈ A(2λ ,ρ /2) and θ ∈ J := [−4π, 4π]+ i[−qh/6, qh/6] 1/2,L q qh/6 (8.242)S(θ , z) = g(ϕ, H(z))dϕ. [0,θ] We notice that by Cauchy’s Formula, (8.237)and (8.238) S(θ + 2π, z) − S(θ , z) = g(ϕ, H(z))dϕ [θ,θ+2π] 2π = g(ϕ, H(z))dϕ 0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 75 = 2π(H(z)) = 2π z hence : (θ , z) → S(θ , z) − θ z deﬁnes a holomorphic function on T × A(2λ ,ρ /2). Moreover, from (8.242), qh/6 1/2,L q (8.236)and (8.239) one can write S(θ , z) = g(ϕ, H(z))dϕ −1/2 = (ϕ) H(z)(1 + g(ϕ, H(z)))dϕ −1/2 = γθ H(z) + (ϕ) H(z)g ˚(ϕ, H(z))dϕ −1/2 = θ z(1 + H(z)) + (ϕ) H(z)g ˚(ϕ, H(z))dϕ and we see that −2 L (1 + qh/6). T ×A(2λ ,ρ /2) qh/6 1/2,L q 1/4 Deﬁne U = T × A(2Lε + δ, ρ /2 − δ) and note that by (8.234) one has L,δ qh/6−δ q 1/4 λ ≤ Lε so that U ⊂ T × A(2λ ,ρ /2) and 1/2,L L,0 qh/6 1/2,L q −2 qL . L,0 By Cauchy’s estimates −2 (8.243) 2 q(δL) . C (U ) L,δ 1/16 Let us choose, δ = ε and −7/32 (8.244)L = ε . 1/4 (−7/32)+(8/32) 1/32 (7/16)−(2/16) 5/16 −2 −2 −2 −3 We then have Lε = ε = ε ,L δ = ε = ε ,L δ = 1 1 1 1 1 (7/16)−(3/16) 1/4 ε = ε hence 1 1 1/4 1/32 (8.245) qε . C (T ×A(2ε ,ρ /2)) qh/6 q 1/32 Using Lemma 2.2 and Lemma 4.4, we see that (, T × A(2ε ,ρ /2)) has qh/6 q 2 Wh aC , σ -symmetric Whitney extension such that −1 1/32 −1 (8.246) W = f , W = f Wh ∈ Symp(T × A(3ε ,ρ /3)) Wh qh/7 q 76 RAPHAËL KRIKORIAN −4/32 1/4 and (W − id 1 ε ε ) 1 1 1/8 (8.247) W − id 1 qε . On the other hand taking the derivative of (8.242)wehave (8.248) ∂ S(θ , z) = g(θ , H(z)) and so S is a solution of the Hamilton-Jacobi equation ∂ S (8.249) (θ , (θ , z)) = (θ , g(θ , H(z))) ∂θ (8.250) = H (z)(by (8.235)). −1 Hence, the exact symplectic change of variable W = f ∂ S −1 w = = w + ∂ (θ, z) ∂θ (8.251) W = f : (θ , w) → (ϕ, z) ⇐⇒ ∂ S ϕ = = θ + ∂ (θ, z) ∂ z conjugates to since from (4.82) (θ ,w) H(z) −1 ◦ W = H ⇐⇒ W ◦ ◦ W = . This concludes the proof. 8.6. Consequences on ◦ f .— Let G : (θ , r) → (θ , r + e (θ )) be the diffeomor- F 0 phism introduced in Lemma 8.3 and (8.252) W = G ◦ W. 1/32 We notice that W ∈ Symp (T × A(3ε ,ρ /3)) and that its image contains G(T × qh/7 q qh/7 σ 1 1/32 A(3ε ,ρ /3)) (see (8.246)); from (8.247)and (8.228)wehave 1/8 (8.253) W − id qε . Corollary 8.8. — One has −1 (8.254) W ◦ ◦ f res ◦ W = 2 ◦ f vf F H F with vf 1/4 1/32 (8.255) F exp(−1/(qρ) )F . h,D(0,ρ) T ×A(4ε ,ρ /4)) qh/8 q 1 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 77 Proof. — Recall that from (8.219) and the deﬁnition of (8.222)) ◦ f res = per ◦ f vf F +F F = ◦ f vf . By Lemma 8.3 and Proposition 8.7 −1 −1 G ◦ ◦ G = , W ◦ ◦ W = 2 hence −1 W ◦ ◦ W = 2 and so −1 −1 vf 2 vf vf vf W ◦ ◦ f ◦ W = ◦ f , f = W ◦ f ◦ W F F F F which is (8.254). vf The estimate on F comes from (8.221)and (8.253). 8.7. Proof of Proposition 8.1: existence of Hamilton-Jacobi Normal Form. — Let Wbe the diffeomorphism constructed in Corollary 8.8 (cf. (8.252)). The map (deﬁned in (8.214)) 1/32 1/32 −1 sends ((R + i]− h/8, h/8[)/(2π/q)Z) × A(c; 4q ε , ρ/4) to T × A(4ε , qρ/4). qh/8 1 1 From (8.217), (8.254) one has −1 −1 res W ◦ ◦ ◦ f ◦ ◦ W = 2 ◦ f vf q H F F q hence −1 −1 −1 res ( ◦ W ◦ ) ◦ ◦ f ◦ ( ◦ W ◦ ) q q q F q −1 −1 = ( ◦ 2 ◦ ) ◦ ( ◦ f vf ◦ ). H q F q q q −1 −1 Let W, and f be lifts by j (deﬁned in (8.213)) of ◦ W ◦ , ◦ 2 ◦ ˚ HJ ˚ vf q q H q F q q −1 res res and ◦ f vf ◦ . Since ◦ f is a lift by j of ◦ f one has for some m ∈ Z q q F F F (0 ≤ m ≤ q − 1) −1 res W ◦ ◦ f ◦ W = ◦ HJ ◦ f vf ˚ ˚ 2π(m/q)r F F where HJ −2 2 vf vf ˚ ˚ (8.256) (r) = q H (q(r − c)), F = O(F ). If we deﬁne −1 cor cor (8.257) f cor = W ◦ f cor ◦ W, F = O(F ) F F HJ (8.258) g = g ◦ W (g from (8.211)) RNF RNF 78 RAPHAËL KRIKORIAN one has from (8.211) (note that W commutes with ) 2π(p/q)r HJ −1 HJ −1 per (8.259) (g ) ◦ ◦ f ◦ g = ◦ W ◦ ◦ f ◦ W ◦ f cor F 2π(p/q)r F = ◦ ◦ HJ ◦ f vf ◦ f cor ˚ ˚ 2π(p/q)r 2π(m/q)r F HJ HJ =: ◦ f with (see (8.256), (8.255), (8.212)) 1/32 HJ −1 (8.260) ∈ O (A(c; 5q ε , ρ/5)), HJ −2 2 (8.261) (r) = 2π((p + m)/q)r + q H (q(r − c)) 1/32 HJ vf cor vf cor −1 ˚ ˚ (8.262) F = F + F + O (F , F ) ∈ O (T × A(c; 5q ε , ρ/5)) 2 σ h/9 HJ 1/4 (8.263) F exp(−1/(qρ) )ε. −1 With a slight abuse of notation, we can write W = ◦ W◦ and using (8.258), (8.252) and the deﬁnition of W(cf. Proposition 8.7) we can write 1/32 HJ −1 −1 g = g ◦ ◦ (G ◦ W) ◦ ∈ Symp (T × A(c; 5q ε , ρ/5)). RNF q h/9 q σ The last inequality of (8.212)and (8.253) show that (remember (8.230)) 1/8 HJ 1− 1/8 −1 (8.264) g − id 1 qε + qε qε (ε qρ ε). C 1 HJ Note that since g − id 1 , f − id (cf. (8.203)) and f HJ − id (cf. (8.263)) are C F F 1/q, the conjugation relation (8.259) shows that the integer m appearing in (8.261) must be equal to 0. Hence, HJ −2 2 (8.265) (r) = 2π(p/q)r + q H (q(r − c)). HJ Let us now check that one can choose in O (D D) which satisﬁes a (2A, 2B)- twist condition. Indeed, from (8.239) and Cauchy’s inequality (recall our choice (8.244) −7/32 L = ε ) we see that −2 2 −2 2 q H (q(r − c)) − γ (r − c) 1/32 3 −1 C (T ×A(c;6q ε ,ρ/6)) h/9 −3/32 7/16 11/32 qε ε ≤ qε . 1 1 1 −6/32 11/32 5/32 We now apply Lemma 2.2: since ε × ε ε , there exists a C σ -sym- 1 1 1 1/7 1/8 3 −2 2 metric Whitney extension with C -norm less that qε < ε for (q H (q(r − c)) − 1/32 −2 2 −1 1/8 γ (r − c) , T × A(c; 6q ε , ρ/6)). Using (8.265) and the inequality ε h/9 HJ (1/2) min(A, B) (cf. (8.203)) we see that has a Whitney extension (that we still de- HJ note )suchthat HJ (8.266) satisﬁes a (2A, 2B) − twist condition. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 79 We can now conclude the proof of Proposition 8.1. We deﬁne the disk Dof Propo- sition 8.1 as (cf. (8.230), (8.228)) 1/33 D = D(q c, ρ) q ⊂ D(c, ε ) (8.267) 1/32 1/32 1/33 −1 q c = c, ρ q= 6ε = 6e ≤ ε (ε qρ ε) 1 0 1 C (T) and the disk D can be taken to be (recall |c − c| ε) (8.268) D = D(c, ρ/6) = D(c, ρ) . With these notations, the set of conclusions (8.207)–(8.210) are consequences of (8.266), (8.259), (8.264)and (8.263). 8.8. Extending the linearizing map inside the hole. — In general the previously deﬁned maps g,, H are not holomorphically deﬁned on a whole disk but rather on an annulus with 1/32 inner disk of radius 6ε where ε =e 0 . In this subsection we quantify to which 1 1 C (T) extent the domains of holomorphy of these maps can be extended if one knows that the HJ frequency map coincides on this annulus with a holomorphic function deﬁned on a disk (containing the annulus). Notation 8.9. — In the following we denote by C(0, t) the circle of center 0 and radius t > 0. Proposition 8.10. — If there exists a holomorphic function deﬁned on D(0,ρ ) such that (8.269) − H ≤ ν C(0,ρ /2) then (1/6)− ε =e 0 ν . 1 1 C (T) We prove this proposition in Section 8.8.2. We now take s = 0.(cf. (8.231)) By (8.236)for z ∈ A(λ/2,λ/4), |g(θ , z)| compares to λ andthusfrom(8.235)and (8.227) 2 2 3 z = (θ) g(θ , z) − e (θ ) + O(g ) 1 80 RAPHAËL KRIKORIAN so that 1/2 2 3 (8.270) g(θ , z) = z / (θ ) + e (·) + O(g ) 1/2 2 2 (8.271) = (z / (θ )) + e (θ ) + O(λ ). Let’s introduce 1/2 (8.272) g(θ , z) = + e (θ ) (θ) 2π −1 −1 (8.273) (·) = (2π) g(θ ,·)dθ, H = where the inverse is with respect to composition. The functions and Hare deﬁned on 1/2 {z ∈ C, Lε < |z|} for some ﬁxed L 1, independent of ε , satisfying −1/2 (8.274)L ≤ (ρ /2)ε (wetakehere s = 0, cf. (8.231)). 8.8.1. Computation of a residue. 1/2 Lemma 8.11. — For any circle C(0, t) centered at 0 with Lε < t <ρ /2 one has 1 dθ 2 3/2 2 zH(z) dz = (γ /4) (θ) e (θ ) 2π i 2π C(0,t) T 2π −1 −1/2 where γ = (2π) (θ) dθ . Proof. — We compute the expansion of g(θ ,·) (cf. (8.272)) into Laurent series: on 1/2 C D(0, Lε ): 1/2 1/2 −2 g(θ , z) = (z/ (θ ) ) 1 + (θ)e (θ )z 1 1 1/2 −2 2 −4 −6 = (z/ (θ ) ) 1+ (θ)e (θ )z − ( (θ )e (θ )) z +O(z ) 1 1 2 8 z 1 1 1/2 −1 3/2 2 −3 −5 = + (θ) e (θ )z − (θ) e (θ ) z + O(z ). 1 1 1/2 (θ) 2 8 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 81 2π −1 As a consequence since (z) = (2π) g(θ , z)dθ we have with the notation γ = 2π −1 −1/2 (2π) (θ) dθ the identity −1 −3 −5 (z) = γ(z + a z + a z ) + O(z ) −1 −3 where 2π −1 −1 1/2 a = γ (1/2)(2π) (θ) e (θ )dθ −1 1 (8.275) 2π −1 −1 3/2 2 a = γ (−1/8)(2π) (θ) e (θ ) dθ. −3 1 By our choice (8.229)wehave a = 0 and we can thus write −1 (8.276) = ◦ (id + u) where z = γ z and −3 −5 u(z) = a z + O(z ). −3 If v is deﬁned by (id + u) ◦ (id + v) = id we have −3 −4 v(z)=−a z + O(z ) −3 and therefore 2 −3 −4 2 (8.277) (z + v(z)) = (z − a z + O(z )) −3 2 −2 −3 = z − 2a z + O(z ). −3 Now since H is the inverse for the composition of (cf. (8.273)), z = ( ◦ H)(z), −1 −1 −1 we have by (8.276) H = (id + u) ◦ = (id + v) ◦ and we get by (8.277) γ γ 2 −2 2 2 −2 −3 H(z) = γ z − 2a γ z + O(z ) −3 and thus 2 −2 3 2 −1 −2 zH(z) = γ z − 2a γ z + O(z ). −3 82 RAPHAËL KRIKORIAN 1/2 HencebyCauchy’sformula and(8.275), for any circle C(0, t),Lε < t <ρ /2: 2 2 zH(z) dz=−2a γ −3 2π i C(0,t) dθ 3/2 2 = (γ /4) (θ) e (θ ) . 2π 8.8.2. Proof of Proposition 8.10. 1/2 Lemma 8.12. — Let Lε ≤ λ<ρ /2, L 1 (independent of ε ). One has for z ∈ q 1 A(λ/4,λ/2) 2 2 3 (8.278) |H(z) − H(z) | λ . Proof. — For z ∈ A(λ/4,λ/2), θ ∈ T one has by (8.271), (8.272) |g(θ , z) − g(θ , z)| λ so (cf. (8.237), (8.273)) (8.279) |(z) − (z)| λ . On the other hand, from Lemma L.1 g(θ , z) − g(θ , z ) 2 2 −3/L 2/L e ≤ ≤ e z − z hence (z) − (z ) 2 2 −3/L 2/L (8.280) e ≤ ≤ e . z − z Since z = (H(z)) = (H(z)) and H(z), H(z) z (cf. (8.239)), one has from (8.279) |(H(z)) − (H(z))| λ and so from (8.280) |H(z) − H(z)| λ . Since from (8.239) |H(z) + H(z)| λ we thus have 2 2 3 |H(z) − H(z) | λ . ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 83 We recall that ε =e 0 . The function − H satisﬁes (cf. (8.269), (8.239)) 1 1 C (T) 2 2 − H ν, − H 1/2 1. C(0,ρ /2) C(0,Lε ) Let M > 5and 1/2 1/2 1/M 1−1/M 1−1/M (8.281) λ := (ρ /2) (Lε ) ≤ (Lε ) M q 1 1 (we can assume ρ ≤ 1). By the Three Circles Theorem, 2 1/M − H ν . C(0,λ ) Lemma 8.12 tells us that 2 1/M 3 − H ν + λ C(0,λ ) M M hence for any z in the circle C(0,λ ) 2 1/M 3 |z(z) − zH (z)| λ (ν + λ ) and 2 2 1/M 3 (z(z) − zH (z))dz λ (ν + λ ). M M 2π i C(0,λ ) Since z → z(z) is holomorphic on D(0, 2λ ), z(z)dz = 0 and by Lemma 8.11 C(0,λ ) we get 3/2 2 2 1/M 3 (θ )e (θ ) dθ λ (ν + λ ). M M Since (θ) 1 this gives 2 2 1/M 3 e (θ ) dθ λ (ν + λ ) M M hence remembering (8.281) 5/2 1/(2M) e 2 λ ν + λ 1 L (T) M (1/2)(1−1/M) (5/4)(1−1/M) (1−1/M) 1/(2M) (5/2)(1−1/M) L ε ν + L ε . 1 1 If we deﬁne (5/4)(1−1/M)−1 (1/2)(1−1/M) (5/2)(1−1/M) (1−1/M) 1/(2M) δ = L ε ,μ = L ε ν M M 1 1 −1 this can be written (recall that e 0 = ε qρ ε)for some C > 0 1 C (T) 1 e 2 ≤ Cδ e 0 + Cμ 1 L (T) M 1 C (T) M 84 RAPHAËL KRIKORIAN and we are in position to apply Lemma M.2 (our choice M > 5 implies that for some 2β β> 0, δ ≤ ε 1): −1 2 −1 ε =e ≤ (μ /δ ) + Ch exp(−h/(Cδ ))qρ ε 1 1 C (T) M M (μ /δ ) + exp(−(1/ε ))(β> 0) M M 1 (μ /δ ) + (1/2)ε M M 1 which gives 1−((3/4)(1−1/M)) −(3/2)(1−1/M) 1/(2M) ε L ε ν or equivalently (3/4)(1−1/M) −(3/2)(1−1/M) 1/(2M) ε L ν and taking M = 5+, one ﬁnally gets: −(2−) (1/6)− ε L ν (1/6)− ≤ ν . This completes the proof of Proposition 8.10. 8.9. Proof of Proposition 8.1: the Extension Property. — From (8.265) we see that if there exists a holomorphic function deﬁned on Dsuch that HJ − ν (4/5)D(1/5)D there exists a holomorphic function deﬁned on D(0,ρ ) (recall that ρ = qρ/3 cf. q q (8.226)) such that − H ν C(0,ρ /2) and thus by Proposition 8.10 (1/6)− ε =e 0 ν . 1 1 C (T) 1/200 Now (8.267) shows that the conclusion of Proposition 8.1 holds with D = D(c,ν ). 9. Comparison Principle for Normal Forms In this section, if 0 ≤ ρ <ρ ,wedenoteby A(c; ρ ,ρ ) the annulus {z ∈ C,ρ ≤ 1 2 1 2 1 |z − c| <ρ } (it is thus the disk D(c,ρ ) if ρ = 0). 2 2 1 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 85 Proposition 9.1 ((AA) Case).— There exist positive constants C, a , a for which the following 4 5 holds. Let 0 <ρ <ρ (resp. ρ = 0 <ρ ), ε, ν > 0 and for j = 1, 2, ∈ O (A(c; ρ ,ρ )), 1 2 1 2 j σ 1 2 F ∈ O (W ),g ∈ Symp (W ) such that: , satisfy an (A, B)-twist condi- j σ h,A(c;ρ ,ρ ) j h,A(c;ρ ,ρ ) 1 2 1 2 σ 1 2 tion (2.59)and −1 g − id ≤ ε< C h j C (9.282) F ≤ ν j W h,A(c;ρ ,ρ ) 1 2 and on g (A(c; ρ ,ρ )) ∩ g (A(c; ρ ,ρ )) one has 1 1 2 2 1 2 −1 −1 g ◦ ◦ f ◦ g = g ◦ ◦ f ◦ g . 1 F 2 F 1 1 1 2 2 2 Then, if δ> 0 satisﬁes − a (9.283) Cε ≤ δ/4 <(ρ − ρ ) and Cδ ν< 1, 2 1 there exists γ ∈ R, |γ|≤ Cε such that one has − a ∂ (·+ γ) − ∂ ≤ Cδ ν. 1 2 A(c;ρ +δ,ρ −δ) 1 2 (9.284) −a (resp. ∂ (·+ γ) − ∂ ≤ Cδ ν.) 1 2 D(c,ρ −δ) Furthermore, if g and g are exact symplectic on M , one can choose γ = 0. 1 2 R Proof. — We only treat the case ρ > 0(the case ρ = 0 is done similarly). 1 1 From (9.282) we see that there exists C > 0 such that one has on W := T × 1 h−Cε A(c; ρ + Cε, ρ − Cε) 1 2 (9.285) g ◦ = ◦ g ◦ f 1 2 where −1 g := g ◦ g ∈ Symp (W ) 1 1 2 σ −1 −1 F ∈ O(W ), f := g ◦ f ◦ g ◦ f , F ν. 1 F F W 2 F 1 We write (9.286) g(θ , r) = (θ + u(θ , r), r + v(θ , r)) −1 and we introduce the notations ω = ∂ , i = 1, 2 (we drop the usual factor (2π) ). We i i have g ◦ (θ , r) = (θ + ω (r) + u(θ + ω (r), r), r + v(θ + ω (r), r)) 1 1 1 1 86 RAPHAËL KRIKORIAN and ◦ g = (θ + u(θ , r) + ω (r + v(θ , r)), r + v(θ , r)) We thus have on W := T × A(c; ρ + Cε + δ, ρ − Cε − δ) 2 h−Cε−Bρ −δ 1 2 ω (r + v(θ , r)) − ω (r) = I + u(θ + ω (r), r) − u(θ , r) 2 1 1 (9.287) v(θ + ω (r), r) − v(θ , r) = II −b with max(I ,II ) = O(δ ν). We observe that from the twist assumption on W W 1 2 2 there exists a set R ⊂ A(c; ρ + Cε + δ, ρ − Cε − δ) of Lebesgue measure δ , which 1 2 is a countable union of disks centered on the real axis, such that one has for any r ∈ A(c; ρ + Cε + δ, ρ − Cε − δ) Rand any k ∈ Z 1 2 l δ (9.288)min|ω (r) − 2π |≥ l∈Z k k so that the second identity in (9.287)gives forany r ∈ A(c; ρ + Cε + δ, ρ − Cε − δ) R 1 2 the following inequality on T (where h = h − Cε − Bρ ) h −2δ 1 2 −3 −b (9.289) v(·, r) − v(θ , r)dθ δ δ ν. h −2δ We now notice that there exists 0 ≤ t ≤ δ such that R∩ ∂ A(c; ρ + Cε + δ + t,ρ − Cε − 1 2 δ − t) =∅. The maximum principle applied, for any ϕ ∈ T , to the holomorphic h −2δ function v(ϕ,·) − v(θ ,·)dθ deﬁned on A(c; ρ + Cε + δ + t,ρ − Cε − δ − t) shows 1 2 that (9.289) holds for any r ∈ A := A(c; ρ + Cε + 2δ, ρ − Cε − 2δ).Wethushave 2δ 1 2 −(4+b) (9.290) ∂ v = O(δ ν). θ h −3δ,A 1 3δ Taking the ∂ derivative of the ﬁrst line of (9.287) and using the previous inequality show that (from now on the value of b may change from line to line) −b ∂ u(θ + ω (r), r) − ∂ u(θ , r) = O(δ ν). θ 1 θ By the same argument used to establish (9.290)weget −b (9.291) ∂ u = O(δ ν) θ h −4δ,A 1 4δ (wehaveused thefactthat ∂ u(θ , r)dθ = 0). Since g is symplectic on W ,det Dg(θ , r) ≡ θ 1 1hence (1 + ∂ u(θ , r))(1 + ∂ v(θ , r)) − ∂ u(θ , r)∂ v(θ , r) = 1 θ r r θ andinviewof(9.290), (9.291) −b ∂ v = O(δ ν) r h −4δ,A 1 4δ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 87 which combined with (9.290) implies, −b (9.292) v − γ = O(δ ν), γ = v(0, 0) ∈ R. h −4δ,A 1 4δ The ﬁrst equation of (9.287) implies that −b ω (·+ γ) − ω (·) = O(δ ν) 2 1 A 4δ which is the ﬁrst conclusion of the Proposition (9.284). If g and g are exact symplectic, g is also exact symplectic and one can write g = f 1 2 Z for some Z = O (g − id) which means g(θ , r) = (ϕ, R) if and only if r = R + ∂ Z(θ , R), 1 θ ϕ = θ + ∂ Z(θ , R). In particular r = r + v(θ , r) + ∂ Z(θ , r + v(θ , r)) and since Z(θ , r + v(θ , r)) = ∂ Z(θ , r + v(θ , r)) + ∂ Z(θ , r + v(θ , r)∂ v(θ , r) θ R θ dθ we get from (9.290) −b v(θ , r)=− Z(θ , r + v(θ , r)) + O(δ ν) dθ which after integration in θ yields −b v(θ , r)dθ = O(δ ν). We can now conclude from (9.292)that −b γ = O(δ ν). In particular, taking γ = 0 does not affect the estimate (9.284). Proposition 9.2 ((CC)-Case).— Under the assumptions of the previous Proposition 9.1: (1) If c = 0, ρ = ρ , ρ = 0 and g , g are exact symplectic then 2 1 1 2 −a ∂ (·) − ∂ (·) ≤ Cδ ν. 1 2 D(0,ρ −δ) (2) If ρ < |c|/4 then all the conclusions of the previous Proposition 9.1 are valid. Proof. — The proof of Item (2) follows from Item (2) of Lemma K.1 of the Ap- pendix applied to Proposition 9.1. 88 RAPHAËL KRIKORIAN So we concentrate on the proof of Item (1), c = 0, ρ = ρ , ρ = 0. We use the 2 1 −1 symplectic change of coordinates of Section K (θ , r) = ψ (z,w), ± −2h 2h ψ : T × (0,ρ) → W ± ∩{e < |z|/|w| < e } ± h h, (0,ρ) α α −1 −1 where α< π/10. Setting g = ψ ◦ g ◦ ψ , g = g ◦ g , g (θ , r) = (θ + u (θ , r), r + j,± j ± ± 1,± ± ± ± 2,± v (θ , r)) we are then reduced to the preceding situation where g is replaced by g ,the ± ± annulus A(c ; ρ ,ρ ) is replaced by the angular sector (ρ − 4δ) and h by h − 4δ, 2 1 2 α+4δ so that (9.287) holds on T × (ρ − 4δ). Like in the previous case, one can ﬁnd h−4δ α−4δ 0 ≤ t ≤ δ such that the Diophantine condition (9.288) holds for any r ∈ (0; t,ρ − α+4δ 4δ − t) := (ρ − 4δ) ∩ A(0; t,ρ − 4δ − t). Still by the Maximum Principle (9.290) α+4δ holds on T × (0; t,ρ − 5δ − t) with v replaced by v and one can conclude as h−5δ ± α+5δ we’ve done before that (9.291) holds with u replaced by u as well. Finally this gives the existence of γ = v (0, 0) ∈ R such that on (0; t,ρ − 5δ − t) ± ± α+5δ −b (9.293) ω (·+ γ ) − ω (·) = O(δ ν). 2 ± 1 A 4δ Now, if g and g areexact symplectic thesameistrue for g , g (cf. Remark 4.2)and 1 2 1,± 2,± hence g is also exact symplectic; we can thus prove, like in the proof of Proposition 9.1, −b that ±γ = O(δ ν). We can hence assume that γ = 0 in equation (9.293). Since α< + − π/10 we deduce that on A(0; t,ρ − 5δ − t) = (0; t,ρ − 5δ − t) ∪ (0; t,ρ − α+5δ α+5δ 5δ − t) one has −b ω (·) − ω (·) = O(δ ν). 2 1 A(0;t,ρ−5δ−t) But ω ,ω ∈ O(D(0,ρ)), hence by the Maximum Principle 1 2 −b ω (·) − ω (·) = O(δ ν). 2 1 D(0,ρ−5δ−t) 10. Adapted Normal Forms: ω Diophantine Recall that for τ ≥ 1, κ> 0 l κ DC(κ, τ )={ω ∈ R, ∀ k ∈ Z , min|ω − |≥ } 0 0 1+τ l∈Z k |k| DC(τ ) = DC(κ, τ ). κ>0 10h 10h Let h > 0, 0 < ρ< 1, A, B ≥ 1, ∈ O (e D(0, ρ)),F ∈ O (e W ) such σ σ h,D(0,ρ) that −1 −1 2 −1 3 (10.294) ∀ r ∈ R, A ≤ (2π) ∂ (r) ≤ A, and (2π) D ≤ B −1 (10.295) ω := (2π) ∂(0) ∈ DC(κ, τ ) ⊂ DC(τ ) 0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 89 (10.296) ∀ 0 <ρ ≤ ρ, F 10h ≤ ρ , e W h,D(0,ρ) (10.297) where m = max(a , a + 4, a , a , 2b + 10) 1,τ 2 3 4 τ (a , a , a , a are the constants appearing in Propositions 6.5, 7.1, 8.1, 9.1 and b is 1,τ 2 3 4 τ deﬁned by (6.152)). We as usual denote ω = (1/2π)∂ (ω(0) = ω ). 10.1. Adapted KAM domains. — We use in this section the notations of Section 7, in particular we denote j a (10.298) ε := maxD F ≤ ρ . h,D 0≤j≤3 Assumption (10.296) allows us to apply Proposition 7.1 on the existence of a KAM Nor- mal Form on the domain W . We can thus deﬁne holed domains U and maps F , 2h,D(0,ρ) n n , g satisfying the conclusions of Proposition 7.1. n m,n (ρ) 10.1.1. Deﬁnition of the domains U .— Let 0 <β 1and μ ∈]1, 1 + 1/τ[ such that (10.299) μ = 1 + (1 − β) ∈]1, 1 + 1/τ[. deﬁn. We deﬁne for ρ< ρ/4 two indices i (ρ), i (ρ) ∈ N as follows: − + (10.300) i (ρ) = max{i ≥ 1, D(0, 2ρ) ∩ U = D(0, 2ρ)} − i deﬁn. and i (ρ) is the unique index such that (10.301) μ μ μ 2 (N ) ≤ N <(4/3) (N ) ≤ N . i (ρ) i (ρ) i (ρ) − + − i (ρ) We also deﬁne ι(ρ) ∈ R by ι(ρ) ∈ R such that (10.302) −ι(ρ) −1/ι(ρ) ρ = (N ) , N = ρ . i (ρ) i (ρ) − − The next lemma shows how N and N compare with ρ . i (ρ) i (ρ) − + Lemma 10.1. — One has −1 −1 (10.303) (1 + 1/τ ) + O(| ln ρ| ) ≤ ι(ρ) ≤ (1 + τ) + O(| ln ρ| ). In particular, −μ/ι(ρ) (10.304)N ρ , i (ρ) + 90 RAPHAËL KRIKORIAN where for ρ 1 1 μ (10.305) − 2β ≤ ≤ 1 − (β/2). τ ι(ρ) Proof. — To prove (10.303)wejusthavetocheck that −(1+τ) −(1+1/τ ) (10.306) (N ) ρ (N ) . i (ρ) i (ρ) − − See the details in Appendix I.1. We shall say that the domains U , i (ρ) ≤ i ≤ i (ρ),are ρ -adapted KAM domains. i − + For t > 0and i (ρ) ≤ i ≤ i (ρ) we deﬁne − + (t) (t) U = U ∩ D(0, t), D (U ) = D(U )={D ∈ D(U ), D∩ D(0, t) = ∅} i t i i i i U being the domains of Proposition 7.1 and where as usual D(U) denotes the holes of the holed domain U (see Section 2.3.1). By (7.193) (j) (t) i−1 −1 U := U ∩ D(0, t) = D(0, t) D(c , s K ), i j,i−1 i l/k j j=1 (k,l)∈E (10.307) i−1 m=j s = e ∈[1, 2] j,i−1 where (j) E ⊂{(k, l) ∈ Z , 0 < k < N , 0≤|l|≤ N },ω (c ) = l/k. j j j j l/k One can in fact in formula (10.307) restrict the union indexed by j to the set j ∈[i (ρ), i− 1]∩ N; cf. Lemma I.1 of Appendix I. (t) One can also describe U by means of its holes: (t) (10.308)U := U ∩ D(0, t) = D(0, t) D D∈D (U ) t i this decomposition being minimal. In particular, if D, D ∈ D (U ) the inclusions D ⊂ D , t i D ⊂ D do not occur. Proposition 10.2. — Let i (ρ) ≤ i < i ≤ i (ρ). − + (1) The holes D ∈ D (U ) are pairwise disjoint. (3/2)ρ i (2) If D ∈ D (U ), D ∈ D (U ) one has either D ∩ D =∅ or D ⊂ D. (3/2)ρ i (3/2)ρ i (3) The number of holes of U intersecting D(0,ρ) satisﬁes (10.309)#{D ∈ D(U ), D ∩ D(0,ρ) = ∅} ρN . (4) Let D ∈ D (U ) and deﬁne ρ i (ρ) i =−1 + min{i : i (ρ) < i ≤ i (ρ), ∃D ∈ D (U ), D ⊂ D}. D − + ρ i (ρ) + ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 91 −1 Then, D is of the form D = D(c , s K ), s ∈[1, 2], c ∈ R,ω (c )∈{l/k,(k, l) ∈ D D D D i D i D E } and one has D ⊂ U . i i D D (5) Let b be deﬁned by (6.152)(where τ is such that (10.295) is satisﬁed). One has (10.310) D(0,ρ ) ⊂ U . i (ρ) Proof. — We refer to Appendix I.2 for the proofs of Items 1, 2 and 4. Proof of Item 3 on the number of holes. — From (10.307)wejusthavetocheck that for N ∈ N 2 2 #{(k, l) ∈ Z , l/k ∈]ω − s,ω + s[, 0 < k < N, 0≤|l|≤ N} sN . 0 0 If (k, l) belongs to the preceding set one has |l − kω | < sNand thus (k, l) belongs to 2 2 2 [−N, N] ∩{(x, y) ∈ R , |x − ω y|≤ sN} a set which has Lebesgue measure sN .We 2 2 2 2 thus have for large N, #(Z ∩[−N, N] ∩{(x, y) ∈ R , |x − ω y|≤ sN} sN . Proof of Item 5, inclusion (10.310). — Recall that b ≥ τ + 1. Since ω is in DC(τ ), τ 0 −(1+τ) for (k, l) ∈ E , j ≤ i (ρ) − 1 one has |l/k − ω | N . Since satisﬁes a (2A, 2B)- j + 0 j (j) −(1+τ) −(1+τ) −1 −1 twist condition (2A) ≤ ∂ω ≤ 2A one has |c | N and because K N j l/k j j j (j) −(1+τ) −(1+τ) −1 −1 (cf. (7.163)) one has |c |− 2K N ≥ C N ,for some C > 0. Now (7.193) l/k j j i (ρ) −(τ+1) −1 shows that U contains a disk D(0, C N ) and we observe that from (10.305), i (ρ) i (ρ) (τ + 1)(μ/ι(ρ)) < τ + 1 ≤ b hence b −1 (τ+1)μ/ι(ρ) (10.311) D(0,ρ ) ⊂ D(0, C ρ ) ⊂ U . i (ρ) 10.1.2. Covering the holes with bigger disks. — Let us deﬁne (compare with (7.163)) ln N N /(ln N ) i i (10.312) K = N K e i i and for any D ∈ D := D (U ) set ρ ρ i (ρ) −1 D = D(c , K ), D ={D, D ∈ D }. D ρ ρ Notice that for any a > 0, ρ 1and i (ρ) ≤ i ≤ i (ρ) one has a − D + 1/a −1 (10.313) ε K |c |/4. D D Indeed, the inequality of the RHS is due to the fact that |c | >ρ (cf. Proposition 10.2, −1 1/(1+τ) Item 5) combined with the fact that N ρ (cf. (10.306)). i (ρ) The inequality of the LHS is a consequence of (7.163). Let us mention that these disks D are the ones on which we shall later perform a Hamilton-Jacobi Normal Form as described in Proposition 8.1. 92 RAPHAËL KRIKORIAN Lemma 10.3. — The elements of D are pairwise disjoint and for any D ∈ D one has ρ ρ D ⊂ (1/10)D ⊂ 6D ⊂ U , D (1/10)D ⊂ U . i i (ρ) D + Proof. — Let D and D be two distinct elements of D .ByProposition 10.2,Item 1, −2 −1 −1 −2 D∩ D =∅ hence from Lemma 7.3,Item 1 |c − c | N . Since K + K N D D i (ρ) i i i (ρ) + D + −1 −1 we get that D(c , K ) ∩ D(c , K )=∅. D D i i Let us now prove 6D ⊂ U .If 6D is not a subset of U one has for some i i D D −1 −1 −2 D ∈ D(U ), (6D) ∩ D = ∅ hence |c − c |≤ 6K + K N . We can apply i D D D i i i (ρ) D + 1/2 Lemma 7.3,Item 1 to deduce |c − c | ε ; but this implies that D ∩ D = ∅,hence D D i (ρ) D = D (we can apply Proposition 10.2,Item 1, since D, D ∈ D ) and by Proposi- (3/2)ρ tion 10.2,Item 4 we obtain D ⊂ U : a contradiction. Let us prove the second inclusion D (1/10)D ⊂ U . If this is not the case then i (ρ) −1 −2 for some D ∈ D(U ) one has D ∩ (D (1/10)D) = ∅ hence |c − c | K N i (ρ) D D i i (ρ) D + which implies as before using Lemma 7.3 that D = D . But since D ⊂ (1/10)D this leads to a contradiction (otherwise D ∩ (D (1/10)D)=∅). Remark 10.1. — Let us mention (this will be useful in the proof of Theorem 12.3) that 1/2 |D ∩ R| ≤ 1. D∈D 10.1.3. No-Screening Property. — Our key proposition is the following. Proposition 10.4. — For any D ∈ D(U ) such that D ∩ D(0,ρ) = ∅ the triple i (ρ) b −1 −1 (U , D (1/10)D, D(0,ρ /2)) is (10b ) | ln ρ| -good (in the sense of Deﬁnition 3.3). i (ρ) τ Proof. — From Remark 3.1 it is enough to prove that for some U ⊂ U con- i (ρ) b b τ τ taining both D(0,ρ ) and D (1/10)D, the triple (U , D (1/10)D, D(0,ρ /2)) is −1 −1 (10b ) | ln ρ| -good. Lemma 10.5. — There exists a constant C > 0 such that for any 1 ≤ s ≤ 4/3, there exists ρ ∈[sρ, sρ + 10Cρ ] such that D(0,ρ ) ∩ U = D(0,ρ ) D. i (ρ) D∈D(U ) i (ρ) D⊂D(0,ρ ) −1 −2 Proof. — From Lemma 7.3 the holes of D(U ) are C N -separated (some i (ρ) + 1 i (ρ) −1 2μ/ι(ρ) C > 0), hence for some C > 0they are C ρ -separated (cf. (10.304)) and because 1 2 −1 2 of (10.305)theyare C ρ -separated for some C > 0. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 93 −1 However, each of these disks has a radius ≤ 2K ρ . Since they are centered i (ρ) on the real line the conclusion follows. From the previous lemma we deduce the existence of a ρ ∈[(5/4)ρ , (4/3)ρ] such that all the holes D ∈ D(U ) of U intersecting D(0,ρ ) are indeed included in i (ρ) i (ρ) + + D(0,ρ ). We then set U = U ∩ D(0,ρ ) = D(0,ρ ) D. i (ρ) D∈D(U ) i (ρ) D⊂D(0,ρ ) b −1 From (10.310)wehave D(0,ρ ) ⊂ U and for any D = D(c , K ) ∈ D(U ) such D i (ρ) i + that D ∩ D(0,ρ) = ∅ one has D ⊂ D(0,(5/6)ρ ): indeed, since D ∩ D(0,ρ) = ∅, −1 −1 4 4 |c | <ρ + K <ρ + ρ hence |c |+ K <ρ + 2ρ <(5/6)ρ . On the other D D i i D D hand, from Lemma 10.3 D (1/10)D ⊂ U (D ⊂ D(0,ρ )). In this situation we can −1 −1 apply Corollary 3.4 with U = U ,B = D(0,ρ /2), d = K , ε = 2K : the triple i i i i D D (U , D (1/10)D, D(0,ρ /2)) is A-good with ln(6/5) (10.314)A = − (I) b | ln ρ| where i (ρ)−1 −1 ln(K /(20ρ )) (I) := #C (ρ) −1 ln(2K /(ρ )) i=i (ρ) with C (ρ) = #{D ∈ D(U ), D ∩ D(0,ρ) = ∅, i = i}. i i (ρ) D From (10.309)ofProposition 10.2,(10.302), (10.303), (10.312), (7.163) one has i (ρ)−1 ι(ρ) + −1 ln(K N /30) i (ρ) 2 − (I) ≤ ρ N ι(ρ) −1 ln(2K N ) i i (ρ) i=i (ρ) − i (ρ)−1 ι(ρ) −(ln N ) + ln(N /30) i (ρ) ≤ ρ N ι(ρ) −(1/(2(a + 2)))hN /(ln N ) + ln(N /2) 0 i i i (ρ) i=i (ρ) − i (ρ)−1 1+β/2 ρ (N ) (ρ 1) i β i=i (ρ) and since N is exponentially growing with i, 1+β/2 (I) ρ × (N ) . i (ρ) + 94 RAPHAËL KRIKORIAN D(0,ρ) −1 D = D(c , K ) D(0,ρ /2) R-axis −1 D = D(c , sK ) FIG. 9. — Adapted KAM Normal Forms (ω Diophantine) in the complex r -plane. The triple (ρ) b −1 (U , D (1/10)D, D(0,ρ /2)) is C | ln ρ| -good From (10.304)and (10.305)wethusget 1−(1+β/2)μ/ι(ρ) β /4 (10.315) (I) ρ ≤ ρ and from (10.314), if ρ 1 1 1 ≤A(someC > 0). 10b | ln ρ| 10.2. Coexistence of KAM, BNF and HJ Normal Forms on the adapted KAM domain. Notation 10.6. — If W is a σ -symmetric holed domain, we denote by NF (W ) (resp. h,U σ h,U NF (W )) the set of triples (, F, g) with ∈ O (U), F ∈ O (W ),g ∈ Symp (W ) ex,σ h,U σ σ h,U h,U (resp. g ∈ Symp (W )). h,U ex,σ Proposition 10.7 (Adapted Normal Forms).— Let ∈ O (U) and F ∈ O (W ) satisfy σ σ h,U (10.294), (10.295), (10.296). For any β 1 deﬁne i (ρ), μ and i (ρ) according to (10.300), τ − + (10.299)and (10.301). Then for any ρ 1 the following holds: (KAM): Adapted KAM Normal Form (Proposition 7.1). Let D ∈ D (U ). ρ i (ρ) −1 (10.316) [W ] g ◦ ◦ f ◦ g = ◦ f h,U F 1,i (ρ) F i (ρ) 1,i (ρ) ± i (ρ) i (ρ) ± ± ± ± −1 (10.317) [W ] g ◦ ◦ f ◦ g = ◦ f h,U F i ,i (ρ) F i (ρ) i ,i (ρ) i i D + i (ρ) i (ρ) + D + D D + + −1 (10.318) [W ] g ◦ ◦ f ◦ g = ◦ f . h,U F i (ρ),i F i i (ρ),i i (ρ) i (ρ) − D i i D − D − − D D 1/2 m/2 (10.319) g − id 1 ε ≤ ρ 1,i (ρ) C + ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 95 1/2 (10.320) g − id 1 ≤ ε i ,i (ρ) C D + i (1/τ )−2β (10.321) F exp(−(1/ρ) ). i (ρ) W + (ρ) h,U Note that ( , F , g ) ∈ NF (W ) and ∈ TC(2A, 2B). i i i ex,σ h,U i (HJ): Hamilton-Jacobi Normal Form. (Proposition 8.1). For any D ∈ D (U ) there exists D ⊂ ρ i (ρ) HJ HJ HJ D and ( , F , g ) ∈ NF (W ) such that σ q h/9,DD D D D HJ HJ −1 HJ HJ (10.322) (g ) ◦ ◦ f ◦ g = ◦ f [W ] F q i i F h/9,DD D D D D D D HJ 1/9 (10.323) g − id 1 ε HJ (10.324) ∈ TC(2A, 2A) HJ (10.325) F exp(−(1/ρ)). D h/9,(DD) HJ The triple ( , D, D) satisﬁes the Extension Principle of Proposition 8.1. (BNF): Birkhoff Normal Form (Proposition 6.5): BNF BNF BNF There exists ( , F , g ) ∈ NF (W bτ ) such that ex,σ h,D(0,ρ ) ρ ρ ρ BNF −1 BNF BNF BNF (10.326) (g ) ◦ ◦ f ◦ g = ◦ f ,(W bτ ) F F h,D(0,ρ ) ρ ρ ρ ρ BNF m−10 (10.327) g − id ρ . BNF (10.328) ∈ TC(2A, 2B) BNF 1−β (10.329) F exp(−(1/ρ) ) ρ τ h,D(0,ρ ) Proof. — KAM: This is just the content of Proposition 7.1. For inequality (10.321) we note that from (7.170), (7.163), (10.304), (10.305) F exp(−N /(ln(N )) ) i (ρ) h,U i (ρ) i (ρ) + i (ρ) + + −(μ/ι(ρ)) exp(−ρ ) (1/τ )−2β exp(−(1/ρ) ). −1 HJ: Let D ∈ D (U ) where D = D(c , s K ), ω (c ) = p/q, q ≤ N , p ∧ q = 1, be ρ i (ρ) D D i D i + i D D one of the disks obtained in Proposition 10.2,Item 4. By Lemma 10.3 the disk 6D = −1 −1 D(c , 6K ) is included in U . We observe that 6K < |c |/4(cf. (10.313)). Since D i D i D i D D −1 −1 −8 −1 a min(6K ,|c |/4) = 6K <(Aq) and F ε <(6K ) D i h,6D i i i D D i D D D 96 RAPHAËL KRIKORIAN (the last inequality comes also from (10.313)) condition (8.205), (8.203) are satisﬁed and we can apply Proposition 8.1 on Hamilton-Jacobi Normal Forms to ◦ f on the i i D D −1 domain W ⊂ W with ρ = K : there exists a disk D ⊂ D h,D h,U i i −1 (10.330) D := D(c ,ρ ) ⊂ (1/10)D := D(c ,(1/10)K ) ⊂ U q q D i D D i HJ HJ HJ and ( , F , g ) ∈ NF (W ) satisfying (10.322) σ q h/9,DD D D D HJ 1/8 1/9 (10.331) g − id 1 qε ≤ ε i i D D D HJ 1/4 (10.332) F exp(−(K /N ) ). W i i q D D D h/9,(DD) ln N To obtain inequality (10.325) we observe that since K = N with i (ρ) ≥ i ≥ i (ρ) i + D − D i we get −1+ln N i (ρ) −(K /N ) −N . i i D D i (ρ) Because for ρ small enough −1 + ln N ≥ 4(2 + τ) we get i (ρ) 1/4 2+τ −(K /N ) −N i i D D i (ρ) which yields, using (10.302)and (10.303)(ρ 1) 1/4 (2+τ)/ι(ρ) −(K /N ) < −(1/ρ) i i D D < −(1/ρ). BNF: We observe that D(0,ρ ) ⊂ D(0,ρ) and apply Proposition 6.5 to (, F) on e W (we use the smallness condition (10.296)). h,D(0,ρ) 10.3. Comparision Principle. — We now use the result of Section 9 to show that these various Normal Forms match to some very good order of approximation. Lemma 10.8 (Comparing Adapted Normal Forms).— For any β 1,and ρ 1 τ β BNF (1/τ )−3β (10.333) − b ≤ exp(−(1/ρ) ) i (ρ) (1/2)D(0,ρ ) + ρ −2 and for any D ∈ D there exists γ ≤ K ρ D HJ (1/τ )−3β (10.334) − (·+ γ ) ≤ exp(−(1/ρ) ). i (ρ) D (4/5)D(1/5)D Proof. — 1) Proof of (10.333). From (10.326), (10.316) and the fact that W b ⊂ W b ∩ W τ τ h,D(0,ρ ) h,D(0,ρ ) h,U i (ρ) + ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 97 BNF one has on g (W b ) ∩ g (W b ) τ τ 1,i (ρ) h,D(0,ρ ) h,D(0,ρ ) + ρ −1 BNF BNF −1 g ◦ ◦ f ◦ (g ) = g ◦ BNF ◦ f BNF ◦ (g ) . 1,i (ρ) F 1,i (ρ) F + i (ρ) i (ρ) + ρ ρ + + ρ ρ b b τ τ We can then apply Propositions 9.1–9.2 with ρ = ρ , ρ = 0, δ = ρ /2, ε = 2 1 min(m/2,m−10) (1/τ )−2β ρ , ν = exp(−(1/ρ) ) because from (10.329), (10.327), (10.319), (10.321) one sees that condition (9.283)reads min(m/2,m−10) b b b −a (1/τ )−2β τ τ τ 4 Cρ ≤ ρ /4 <ρ and C(ρ /2) exp−(1/ρ) )< 1 BNF and is satisﬁed for ρ 1(cf. (10.297)). Since g and g are exact symplec- 1,i (ρ) BNF −(b +1)a (1/τ )−2β τ 5 tic we then get − τ ≤ Cρ exp(−(1/ρ) ) which is i (ρ) D(0,(1/2)ρ ) (1/τ )−3β ≤ exp(−(1/ρ) ) if ρ is small enough. 2) Proof of (10.334). Similarly, from (10.317), (10.322) one has on the set HJ g (W ) ∩ g (W ) i ,i (ρ) h/9,D(1/5)D D + h/9,D(1/5)D HJ HJ −1 −1 HJ HJ g ◦ ◦ f ◦ (g ) = g ◦ ◦ f ◦ (g ) i ,i (ρ) F i ,i (ρ) D + i (ρ) i (ρ) D + D F D + + D D andfrom(10.320)(10.321), (10.323), (10.325), we see that Propositions 9.1–9.2 apply 1/9 −1 −1 −1 with c = c , ε = ε , δ = K /20, ρ = (1/10)K , ρ = K < |c |/4 since condition i 1 2 i D i i i i D D D D D (9.283) is implied by 1/9 −1 −1 a (1/τ )−2β Cε ≤ K /80 < K /10 and C(20K ) exp(−(1/ρ) )< 1 i i i D D D which is satisﬁed (cf. (7.163), (10.312)) if ρ is small enough. We then get for some γ ∈ R, 1/9 −2 |γ | Cε ≤ K (cf. (10.313)) that on the annulus (4/5)D (1/5)D one has | − D i (ρ) i i + D D HJ (1/τ )−3β (·+ γ )|≤ exp(−(1/ρ) ). 11. Adapted Normal Forms: ω Liouvillian (CC case) 10h 10h Let h > 0, 0 < ρ< 1, A, B ≥ 1, ∈ O (e D(0, ρ)),F ∈ O (e W ) such σ σ h,D(0,ρ) that −1 −1 2 −1 3 (11.335) ∀ r ∈ R, A ≤ (2π) ∂ (r) ≤ A, and (2π) D ≤ B −1 (11.336) ω := (2π) ∂(0) ∈ R Q 10h (11.337) ∀ 0 <ρ ≤ ρ, F ≤ ρ , e W h,D(0,ρ) where (11.338) m = 4 + max(a , 2000Aa , a , a ) 1 2 3 4 (a , a , a , a are the constants appearing in Propositions 6.4, 7.1, 8.1, 9.1). 1 2 3 4 98 RAPHAËL KRIKORIAN Using the notations of Section 6.3.1,let (p /q ) be the sequence of convergents n n n of ω : 1 1 (11.339) ≤|ω − (p /q )|≤ . 0 n n 2q q q q n+1 n n+1 n (11.340) ∀ 0 < k < q , ∀ l ∈ Z, |ω − (l/k)| > . n 0 2kq We assume that n is large enough and we set 10A (11.341) ρ = ≤ ρ/10. q q n+1 n We introduce j m−3 2000Aa (11.342) ε := maxD F (10ρ ) ≤ ρ . W n 2h,D(0,10ρ ) n 0≤j≤3 11.1. Adapted KAM domains. — Since Condition (7.161) is satisﬁed we can apply Proposition 7.1 (with ρ = 10ρ ) and deﬁne holed domains U , functions ,F , ω n i i i Prop. 7.1 i etc. In particular for 0 < t i−1 (j) −1 U ∩ D(0, t) = D(0, t) D(c , s K ), i j,i−1 j=1 (k,l)∈E l/k j (11.343) i−1 m=j s = e ∈[1, 2] j,i−1 where (j) E ⊂{(k, l) ∈ Z , 0 < k < N , 0≤|l|≤ N },ω (c ) = l/k. j j j j l/k Note that from (11.342) and the deﬁnition (7.163)of K −1 1000A 2(a +2) (11.344)K ≤ ε ≤ ρ . j n Lemma 11.1. — Let j be such that N < q /(10A) and (k, l) ∈ E . j n+1 j (1) If (k, l) ∈ Z(q , p ) one has l/k = p /q and n n n n −1 (2A) (2A) (j) 2 −1 (11.345) (40A ) ρ ≤ ≤|c |≤ ≤ ρ /5. n n p /q n n 2q q q q n+1 n n+1 n (2) If (k, l)/∈ Z(q , p ) n n (j) (11.346) |c |≥ 4ρ . l/k Proof. — Item 1 comes from (11.339) and the twist condition (7.166). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 99 To prove Item 2 we observe that if (k, l)/∈ Z(q , p ) n n l l p p 1 1 99A n n |ω − |≥| − |−|ω − |≥ − ≥ ≥ 9Aρ 0 0 n k k q q kq q q q q n n n n n+1 n n+1 (j) and from the twist condition (7.166)weget |c |≥ 4ρ . l/k ∗ − For n ∈ N deﬁne i as the unique index i such that (11.347)N − ≤ q < N − i −1 i n n and i as the unique index (see the deﬁnition of the sequence N in (7.163)) such that (3/4)q q n+1 n+1 (11.348) ≤ N + < . 2 2 (10A) (10A) We deﬁne − − (i ) (i ) −1 n n − + c = c , D := D(c , s K ), D = D(c ,|c |/24) n n n n n p /q p /q i ,i −1 n n n n n n (n) (11.349)U := U + ∩ D(0,ρ ). Note that from (11.345) 2 −1 (11.350) (40A ) ρ ≤|c |≤ ρ /5. n n n Proposition 11.2. — For n large enough, (1) D(0,ρ ) ⊂ U . (n) (2) One has D := D(U )={D }. ρ n (3) One has the following inclusion 6D ⊂ U . −6 (4) One has D(0, q ) ⊂ U + . n+1 i −6 (n) (5) The triple (U , D (1/10)D , D(0, q /2)) is 1/(10| ln ρ |)-good (in the sense of Deﬁ- n n n n+1 nition 3.3). Proof of Item 1.— If j < i and (k, l) ∈ E one has 0 < k < N ≤ q hence from j n i −1 n n (j) (j) −1 (11.346) |c |≥ 4ρ and from (11.344) |c |− 2K ≥ 3ρ . The conclusion then follows n n l/k l/k j from (11.343) applied with i = i . Proof of Item 2.— From Item 1, equality (11.343) can be written i −1 (j) −1 U ∩ D(0,ρ ) = D(0,ρ ) D(c , s K ). n n j,i−1 n l/k j (k,l)∈E j=i j n 100 RAPHAËL KRIKORIAN (i ) We observe that (q , p ) ∈ E and from (11.345), (11.344) one sees that D = D(c , n n i n n p /q n n −1 − + s − + K ) ⊂ D(0,ρ ). More generally, if (k, l) ∈ E , i ≤ j ≤ i − 1and (k, l)/∈ − n j i ,i −1 n n n n (j) 2 −1 Z(q , p ), one has N ≤ q /(10A) and (11.346), (11.344) give that D(0, c , 2K ) ∩ n n j n+1 l/k j (j) −1 − + D(0,ρ )=∅. Since the sets D(0, c , s K ), i ≤ j ≤ i , form a nested decreasing n q ,i −1 p /q n j n n + n n (for the inclusion) sequence of disks one gets U + ∩ D(0,ρ ) = D(0,ρ ) D . n n n Proof of Item 3.— This comes from the fact that |c |+ 6|c |/4 ≤ ρ . n n n −6 Proof of Item 4.— This comes from Item 1 and the fact that |c |−|c |/4 ≥ q as n n n+1 is clear from the LHS inequality of (11.345). −1 Proof of Item 5.— Notice that from (11.345)5 ≤ ρ /|c |≤ 40A and that 2K ≤ n n − 1000A ρ . We use Corollary 3.4; we have to evaluate ln(|c |/(4ρ )) ln(|c |/8ρ ) n n n n I = − −6 −1 ln(q /(2ρ )) ln(2K /ρ ) n − n n+1 ln(20) ln(320A ) ≥ − 7| ln ρ | (1000A − 1)| ln ρ | n n ≥ . 10| ln ρ | 11.2. Adapted Normal Forms. Proposition 11.3. — Let ∈ O (U) and F ∈ O (W ) satisfy (10.294), (10.295), σ σ h,U (10.296). Let 0 <β 1 and n 1 such that (11.351) q ≥ q . n+1 (KAM): Adapted KAM Normal Form ((Proposition 7.1)): One has ( , F , g ∈ NF (W ), i i i ex,σ h,U ∈ TC(2A, 2B) and −1 (11.352) g ◦ ◦ f ◦ g = ◦ f [W ] ± F F h,U 1,i ± ± ± 1,i i i i n n n −1 − + (11.353) g ◦ ◦ f ◦ g = ◦ f [W (n)] − + F F − − i ,i + + h,U n n i ,i n n i i i i n n n n 1/2 m/3 + − + (11.354) g − id 1 ,g − id 1 ε ≤ ρ C C 1,i i ,i n n n n 1−β (11.355) F exp(−q ). i (n) n+1 h,U (HJ): Hamilton-Jacobi Normal Form (Proposition 8.1). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 101 D(0,ρ ) ρ 1/(q q ) n n+1 n D = D(c , c /4) n n n −6 D(0, q ) n+1 R-axis 0 c ρ n n a/3 D = D(c ,ρ ) n n n n −6 FIG. 10. — Adapted KAM Normal Forms (CC Case) in the complex r -plane. The triple (U , U , D(0, q )) is n+1 1/(10| ln ρ |)-good HJ HJ HJ There exists ( , F , g ) ∈ NF (W ) such that n n n h/9,D D n n HJ −1 HJ HJ HJ (11.356) (g ) ◦ ◦ f ◦ g = ◦ f [W ] F q − − n n F h/9,(D D ) n n n n i i n n 1/9 HJ m/9 (11.357) g − id 1 ε ≤ ρ n n HJ (11.358) ∈ TC(2A, 2A) (1/4)−β HJ (11.359) F exp(−q ). n q n+1 h/9,(D D ) n n HJ The triple ( , D, D) satisﬁes the Extension Principle of Proposition 8.1. (BNF): Birkhoff Normal (Proposition 6.4): BNF BNF BNF −6 There exists ( , F , g ) ∈ NF (W ) such that −1 −1 −1 ex,σ h,D(0,q ) q q q n+1 n+1 n+1 n+1 BNF −1 BNF (11.360) (g ) ◦ ◦ f ◦ g = BNF ◦ f BNF [W −6 ] −1 −1 F F h,D(0,q ) q q −1 −1 n+1 n+1 n+1 q q n+1 n+1 −(m−27) BNF (11.361) g − id q −1 −6 n+1 h,D(0,q ) n+1 n+1 BNF (11.362) ∈ TC(2A, 2B) −1 n+1 1−β BNF (11.363) F ≤ exp(−q ). −1 −6 n+1 h,D(0,q ) n+1 n+1 102 RAPHAËL KRIKORIAN Proof. — KAM: This is Proposition 7.1. Inequality (11.355) comes from the corre- + + + + sponding (7.170) ε ≤ exp(−N /(ln N ) ) and the fact that N q . i i i i n+1 n n n n HJ: By Proposition 11.2,Item 3,the disk 6D = D(c ,|c |/4) is included in U . Since n n n 1/8 −1 a − − (6(|c |/24)) <(Aq ) , and F ε <(|c |/4) n n i h,6D i n n n n (the ﬁrst inequality is a consequence of (11.351) and the second of (11.342)and thefact that |c | ρ )(8.205), (8.203) are satisﬁed and we can apply Proposition 8.1 on Hamilton- n n Jacobi Normal Forms to ◦ f on the domain W ⊂ W with ρ =|c |/24: there F h,U n − − h,D − i i i n n n exists a disk D ⊂ D n n (11.364) D := D(c ,ρ ) ⊂ (1/10)D ⊂ D = D(c ,|c |/24) ⊂ U − n q q n n n n D D n n n HJ HJ HJ and ( , F , g ) ∈ NF (W ) such that one has (11.356)and σ q n n n h/9,D D n n HJ 1/8 1/9 (11.365) g − id q ε ≤ ε W n − − D h/9,(D D ) i i n n n n n HJ 1/4 (11.366) F exp(−1/(q |c |/24) ) W n n D h/9,(D D ) n n n −1 and since |c | (q q ) (n 1) n n n+1 β HJ (1/4)−β F exp(−q ). n+1 D h/40,(D D ) n n n 1/10 BNF: Since F ≤ ρ (cf. (11.337)) we can apply Proposition 6.4 on the exis- e W h,D(0,ρ ) n tence of approximate BNF in the CC case (with n + 1 in place of n): for 0 <β 1and BNF BNF BNF n 1: there exists ( , F , g ) ∈ NF (W −6 ) such that β −1 −1 −1 ex,σ h,D(0,q ) q q q n+1 n+1 n+1 n+1 BNF −1 BNF [W −6 ] (g ) ◦ ◦ f ◦ g = BNF ◦ f BNF −1 F −1 h,D(0,q ) F q q n+1 −1 −1 n+1 n+1 q q n+1 n+1 with 1−β BNF F ≤ exp(−q ). −1 W −6 n+1 h,q n+1 n+1 11.3. Comparision Principle. — These various Normal Forms match to some very good order of approximation. Lemma 11.4 (Comparing Adapted Normal Forms).— One has for any β 1,n 1 1−β BNF (11.367) + − −6 exp(−q ) −1 i (1/2)D(0,q ) n+1 n+1 n+1 −m m/2 and there exists γ q |c | such that n n n+1 HJ (1/4)−β (11.368) + − (·+ γ ) exp(−q ) i (4/5)D (1/5)D n+1 n n n n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 103 Proof. — Let us prove estimate (11.367). From (11.360)–(11.352), we see that on BNF g (W −6 ) ∩ g (W −6 ) one has −1 1,i h,D(0,q h,D(0,q n+1 n+1 n+1 BNF BNF −1 −1 BNF BNF + + g ◦ ◦ f ◦ (g ) = g ◦ ◦ f ◦ (g ) . −1 −1 F F 1,i + + 1,i n n q q −1 −1 i i n n n+1 q q n+1 n+1 n+1 1−β −6 −7 We then apply Proposition 9.2 with c = 0, ρ = 0, ρ = q , δ = q , ν = exp(−q ) (cf. 1 2 n+1 n+1 n+1 − min(m/3,m−27) (11.363), (11.355)), ε = q (cf. (11.361), (11.354)) (estimates (9.282)and (9.283) n+1 − min(m/3,m−27) 1/2 −7 −6 7 a are satisﬁed since Cq ≤ q /4 q and q exp(−q ) 1). n+1 n+1 n+1 n+1 n+1 Estimate (11.368) is a consequence of Proposition 9.2 applied to (11.353)and (11.356) −1 HJ HJ −1 g − + ◦ ◦ f ◦ (g − + ) = g ◦ HJ ◦ f HJ ◦ (g ) i ,i + + i ,i n n n n n F n n n i i n n with A(c; ρ ,ρ ) = D (1/10)D , c = c , ρ =|c |/40, ρ =|c |/4, δ =|c |/10, 1 2 n n n 1 n 2 n n n 1/5 m/9 ν = exp(−q ) (cf. (11.359), (11.355)), ε = ρ (cf. (11.354), (11.357)). Estimates n+1 m/9 (9.282)and (9.283) are satisﬁed since Cρ ≤|c |/40 |c |/5(cf. (11.350)) and n n (1/4)− −a −1 C(|c |/10) exp(−q )< 1(recall that |c | (q q ) ). n n n n+1 n+1 12. Estimates on the measure of the set of KAM circles We refer to Section 4.4 for the notations of this section. We observe (W ) := W ∩ M = W := {r ∈ U ∩ R}∩ M . h,U R h,U R U∩R R In particular in the (AA)-case (W ) = W = T × (U ∩ R) and in the (CC*)-case h,U R U∩R 2 2 2 (W ) = W ={(x, y) ∈ R ,(1/2)(x + y ) ∈ U ∩ R }. h,U R U∩R + 12.1. Classical KAM estimates. — We ﬁrst state a variant of the classical KAM the- orem on abundance of invariant circles which is a consequence of Propositions 7.1, 7.2, 7.5 and Remark 7.1 on KAM Normal Forms. In the next theorem the constant a is the one of Proposition 5.5 and the constant a was deﬁned in Section 7 by (7.160). Theorem 12.1. — Let U be a holed domain with disjoint holes D ∈ D(U) such that 1/2 (12.369) |D ∩ R| ≤ 1 D∈D(U) and ∈ O (U) ∩ TC(A, B) (cf. (7.158)) with A, B satisfying (2.60), F ∈ O (W ) σ σ h,U ε := F ≤ d(U) . h,U 104 RAPHAËL KRIKORIAN Then, if f = ◦ f one has 1/(2(a +3)) Leb (W −1/10 L(f , W )) (F ) . M e U∩R U∩R W R h,U Proof. — See Appendix J.2. Notation 12.2. — We deﬁne for ρ> 0, D (0,ρ) = D(0,ρ) ∩ R =] − ρ, ρ[ and 1/2 m (ρ) = Leb (W L(f , D (0, e ρ))). f M D (0,ρ) R R R 12.2. Estimates on the measure of the set of invariant circles: ω Diophantine (AA) or (CC) Case. — We use the notation of Section 10 and assume that (both in the (AA) or (CC)- cases) (10.294), (10.295)(10.296) hold. We denote (12.370) D = D(U ). ρ i (ρ) Theorem 12.3. — For any β> 0, ρ 1 (1/τ )−β (AA)-case m (ρ) exp(−(1/ρ) ) + |D ∩ R|. ◦f D∈D (1/τ )−β (CC) or (CC*)-case m (ρ) exp(−(1/ρ) ) + |D ∩ R |. ◦f + D∈D Moreover, for any D ∈ D one has −β 1+τ (12.371) |D ∩ R| exp(−(1/ρ) ). Proof. — If S ⊂ C we denote S = S ∩ R (if c ∈ R, D (c, t) = D(c, t) ∩ R =]c − R R t, c + t[). 1/4 1/3 Choose (cf. Lemma 10.5) ρ ∈[e ρ, e ρ] (ρ 1) such that (ρ ) U := D(0,ρ ) ∩ U = D(0,ρ ) D i (ρ) D∈D(U ) i (ρ) D⊂D(0,ρ ) hence (ρ ) −1/10 −1/10 (12.372) e D (0,ρ ) ⊂ e U ∪ (1/4)D . R R D∈D HJ Let us denote for short f = ◦ f , f = HJ ◦ f HJ and i F i i D D D HJ L = L(f , W ), L = L(f , W ). (ρ ) q i (ρ) i (ρ) D + + D D D U R R R ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 105 We have from (10.318)(10.322)(10.316)ofProposition 10.7 −1 (12.373) W , g ◦ f ◦ g = f h,U i (ρ) i (ρ),i i i i (ρ),i − − D D D − D HJ HJ HJ −1 (12.374) W ,(g ) ◦ f ◦ g = f h/9,DD D D D D −1 (12.375) W , g ◦ f ◦ g = f h,U i (ρ) i (ρ),i (ρ) i (ρ) i (ρ) i (ρ),i (ρ) − − + + + − + −1 (12.376) W , g ◦ f ◦ g = f . h,U 1,i (ρ) i (ρ) − − i (ρ) 1,i (ρ) − − From Lemma 10.3,Remark 10.1 and estimate (10.321) on the one hand, and estimate HJ (10.325) on the other hand, we see that we can apply Theorem 12.1 to f and f to i (ρ) + D get the following decompositions (12.377)W L ⊂ B , W −1/10 L ⊂ B ∪ E (ρ ) q q −1/10 i (ρ) i (ρ) e D D + + R DD D e U with B ⊂ W ,E = W and −1/10 q q q q DD D D D e D R R R (1/τ )−β/2 (12.378)max Leb (B ), Leb (B ) exp(−(1/ρ) ) M i (ρ) M R + R DD (12.379)Leb (E ) Leb (W ). −1/10 M q M q R D R e D We now introduce HJ (12.380) L := g (L ), L := g ◦ g (L ) i (ρ) i (ρ),i (ρ) i (ρ) i (ρ),i + − + + D − D D D HJ (12.381) B = g (B ), B = g ◦ g (B ), i (ρ) i (ρ),i (ρ) i (ρ) q i (ρ),i q + − + + DD − D D DD HJ (12.382) E = g ◦ g (E ). q q D i (ρ),i D − D Lemma 12.4. — One has −1/10 g (W ) L ⊂ B e D (0,ρ ) i (ρ),i (ρ) R − + with L = L ∪ L B = B ∪ (B ∪ E ). q q i (ρ) D i (ρ) + + DD D D∈D D∈D ρ ρ Proof. — We observe that from (12.372) one has −1/10 (12.383)W ⊂ W (ρ ) ∪ W e D (0,ρ ) −1/10 (1/4)D R R e U D∈D ρ 106 RAPHAËL KRIKORIAN hence g (W −1/10 ) e D (0,ρ ) i (ρ),i (ρ) R − + ⊂ g (W ) ∪ g (W ). (ρ ) −1/10 (1/4)D i (ρ),i (ρ) i (ρ),i (ρ) R − + e U − + D∈D HJ 1/9 Note that by Proposition 10.7 one has max(g − id 1 ,g − id 1 ) ≤ ε i (ρ),i (ρ) C C − + i (ρ) D − −1 2 K (since N ≤ N )hence i D i (ρ) D − HJ g (W ) ⊂ W ⊂ g ◦ g (W −1/10 ) (1/4)D (1/2)D e D i (ρ),i (ρ) R R i (ρ),i R − + − D D which yields g (W −1/10 ) e D (0,ρ ) i (ρ),i (ρ) R − + HJ ⊂ g (W ) ∪ g ◦ g (W ) ∪ E . (ρ ) −1/10 q q −1/10 i (ρ),i (ρ) i (ρ),i e D D D − + e U − D D R R D∈D We then conclude using (12.377) and the notations (12.380). 1/10 Lemma 12.5. — For some G ⊂ W one has L = L(f , G) and e D (0,ρ ) i (ρ) R − (1/τ )−β (12.384)Leb (B) exp(−(1/ρ) ) + Leb (W ). −1/10 M M q R R e D D∈D Proof. — We observe that from (4.86) (12.385) L := g (L ) = L(f , g (W )) (ρ ) i (ρ) i (ρ),i (ρ) i (ρ) i (ρ) i (ρ),i (ρ) + − + + − − + HJ HJ (12.386) L := g ◦ g (L ) = L(f , g ◦ g (W )) D i (ρ),i D i (ρ) i (ρ),i q − D D − − D D D D R R hence, L = L(f , G) i (ρ) with HJ G = g (W ) ∪ g ◦ g (W ) (ρ ) i (ρ),i (ρ) i (ρ),i q − + − D D D D U R R D∈D and clearly G ⊂ W 1/10 . e D (0,ρ ) To get the estimate on the measure of Bwe use (12.378)and (12.381)toget (1/τ )−β/2 Leb (B )) exp(−(1/ρ) ), M i (ρ R + ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 107 and (remember (10.309), (10.304), (10.305)) 2 (1/τ )−β/2 Leb B N exp(−(1/ρ) )) M q R DD i (ρ) D∈D (1/τ )−β exp(−(1/ρ) ); moreover (see (12.379), (12.382)) Leb E Leb (W ). q −1/10 q M M R D R e D D∈D D∈D ρ ρ Summing up these estimates yields the desired inequality on the measure of B. End of the proof of Theorem 12.3. Lemmata 12.4 and 12.5 give 1/10 g (W ) L(f , W ) ⊂ B D (0,ρ ) i (ρ) e D (0,ρ ) i (ρ),i (ρ) R − R − + hence 1/10 (12.387) g ◦ g (W ) g (L(f , W )) ⊂ g (B). D (0,ρ ) i (ρ) e D (0,ρ ) 1,i (ρ) i (ρ),i (ρ) R 1,i (ρ) R 1,i (ρ) − − + − − −1 Since the conjugation relation g ◦ f ◦ g = f holds on W (cf. (12.376)) 1,i (ρ) i (ρ) U ∩R − − 1,i (ρ) i (ρ) − − 1/10 and since W ⊂ W (recall that by deﬁnition (10.300) D(0, 2ρ) ⊂ W e D (0,ρ ) U ∩R h,U R i (ρ) i (ρ) − − and that g − id ρ ) one has by (4.86) 1,i (ρ) L(f , g (W 1/10 )) = g L(f , W 1/10 ). e D (0,ρ ) i (ρ) e D (0,ρ ) 1,i (ρ) R 1,i (ρ) − R − − Equation (12.387) then implies that 1/10 g ◦ g (W ) L(f , g (W )) ⊂ g (B). D (0,ρ ) e D (0,ρ ) 1,i (ρ) i (ρ),i (ρ) R 1,i (ρ) R 1,i (ρ) − − + − − Finally, inclusions W ⊂ g ◦ g (W ) and g (W 1/10 ) ⊂ D (0,ρ) D (0,ρ ) e D (0,ρ ) R 1,i (ρ) i (ρ),i (ρ) R 1,i (ρ) R − − + − W 1/2 yield e D(0,ρ) Leb (W L(f , W 1/2 )) Leb (B). M D (0,ρ) e D(0,ρ) M R R R We conclude by using the estimate (12.384) and the fact that |D ∩ R|, (AA)-case Leb (W ) ≤ −1/10 M q R e D |D ∩ R |, (CC) or (CC*)-case. Proof of estimate (12.371) on the size of the holes. 108 RAPHAËL KRIKORIAN Referring to (12.374)and (8.206)ofProposition 8.1 we see that 1/33 |D| f D h,U where i (ρ) ≤ i ≤ i (ρ).From(7.163) − D + 1/33 1/33 −hN /(33(ln N ) ) i i D D f ≤ ε = e i W i h,U D 1− −N 1− −N i (ρ) e (since i (ρ) ≤ i ) − D hence from (10.302)for any β> 0 (1/ι(ρ))(1−β) |D| exp(−(1/ρ) ). Using (10.303) we then get if ρ 1(D is a disk centered on the real axis) −β 1+τ |D ∩ R| exp(−(1/ρ) ). 12.3. Estimates on the measure of the set of invariant circles: ω Liouvillian, (CC)-Case. — We now assume that (11.335), (11.336)(11.337) hold. Theorem 12.6. — Let ρ = (10A)/(q q ) and assume that q ≥ q .Then, forall n n n+1 n+1 β 1 and n 1 one has 1/4−β m (ρ ) exp(−q )+|(D ∩ R )|. ◦f n n + F n+1 Moreover, 1−β −q (12.388) |(D ∩ R )| e . n + Proof. — The principle of the proof is the same as that of Proposition 12.3 with the HJ ± HJ following modiﬁcations in the notations: we set f = f , f = f and we replace in the i F ± n D n F n D ± − q q proof the indices i (ρ) by i , D, D, Dby D ,D D , i by i , ρ by (4/3)ρ , ρ by 2ρ , ± n n n D n n n n (1/4)− ρ (n) (1/τ )−β U by U and exp(−(1/ρ) ) by exp(−q ). Instead of using the conjugation n+1 relations of Proposition 10.7 (Adapted Normal Forms in the (CC) or (CC*)-case) we use those of Proposition 11.3. Estimate (12.388)isprovedlike(12.371) by noticing that 1/33 1/33 −(1−) f ε ≤ exp(−N ) i W i i h,U D − D n i n and using (11.347). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 109 Remark 12.1. — Note that if the twist condition (11.335)issatisﬁed, then any twist condition TC(A , B) is satisﬁed with A ≥ A. We can thus replace in Theorem 12.6 ρ = (10A)/(q q ) by ρ = (10A )/(q q ) for any ﬁxed A ≥ A(then n has to be chosen n n n+1 n n n+1 larger). 13. Convergent BNF implies small holes 13.1. Case where ω Diophantine in the (AA) of (CC) setting. — We keep here the no- tations of Sections 10 and 12.2, in particular we assume ω is τ -Diophantine and that (10.294), (10.295), (10.296) hold. Lemma 13.1. — If BNF( ◦ f ) converges and is equal to a holomorphic function ∈ O(D(0, 1)) then for all β> 0, ρ 1 and for any D ∈ D β ρ (1/τ )−β (13.389) − exp −(1/ρ) . i (ρ) + D(1/10)D −2 As a corollary, for any D ∈ D and γ ≤ K ρ D HJ (1/τ )−β (13.390) − (·− γ) exp −(1/ρ) . D (4/5)D(1/5)D Proof. — Let us prove inequality (13.389). From (10.333) and Proposition 6.7 one gets (1/τ )−β/2 1−β − b exp(−(1/ρ) ) + exp(−(1/ρ) ) i (ρ) (1/2)D(0,ρ ) (1/τ )−β/2 exp(−(1/ρ) ). (ρ) Since the function − is holomorphic on U and since the triple (U , D i (ρ) i (ρ) + + b −1 −1 (1/10)D, D(0,ρ /2)) is (10b ) | ln ρ| -good, cf. Proposition 10.4,wehavebyDeﬁni- tion 3.3 −1 (1/τ )−β/2 − exp − (10b | ln ρ|) (1/ρ) i (ρ) D(1/10)D τ (1/τ )−β exp(−(1/ρ) ). The inequality (13.390) is then a consequence of (13.389)and (10.334). Corollary 13.2. — If BNF(f ) = then for all β> 0, ρ 1, and any D ∈ D the β ρ radius ρ of the disk D satisﬁes (1/τ )−β ρ exp −(1/ρ) . D 110 RAPHAËL KRIKORIAN Proof. — This results from (13.390) and the Extension Property in Proposition 8.1. Corollary 13.3. — If BNF( ◦ f ) converges then for all β> 0, ρ 1 (recall Nota- F β tion 12.2 for m) (1/τ )−β m (ρ) exp −(1/ρ) . ◦f Proof. — This is a consequence of the previous Corollary 13.2 and of Proposi- 1−2(μ/ι(ρ)) −1 tion 12.3 since #D ρ ρ (cf. (10.309), (10.304), (10.305)). 13.2. Case ω is irrational in the (CC) setting. — The notations here are those of Section 11. In particular we assume that (11.335), (11.336), (11.337) hold. Lemma 13.4. — If BNF( ◦ f ) converges and is equal to ∈ O(D(0, 1)) then for all β 1,n 1 such that q ≥ q β n+1 1−β (13.391) − exp(−q ). D (1/10)D i n+1 n n n −m m/2 As a corollary, for γ q |c | n n n+1 HJ (1/4)−β (13.392) − (·− γ )) exp(−q ). (4/5)D (1/5)D n n n+1 Proof. — Let us prove (13.391). From (11.367) and Proposition 6.7 one gets 1−β/2 − −6 exp(−q ). D(0,q /2) n+1 n+1 (n) (n) Since the function − is holomorphic on U and since the triple (U , D −6 (1/10)D , D(0, q /2)) is 1/(10| ln ρ |)-good (see Proposition 11.2,Item(5)), we have n n n+1 by Deﬁnition 3.3 (remember (11.341)) −1 1−β/2 − exp −(10| ln ρ |) (q ) n n+1 i D (1/10)D n n n 1−β exp(−q ). n+1 The inequality (13.392) is then a consequence of (13.391)and (11.368). Corollary 13.5. — If BNF(f ) = then for any β> 0,n 1 such that q ≥ q ,the β n+1 radius ρ of the disk D satisﬁes q n (1/4)−β ρ exp(−q ). D n+1 n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 111 Proof. — This results from (13.392) and the Extension Property of Proposition 8.1. Corollary 13.6. — If BNF( ◦ f ) converges, then for any β> 0, A ≥ A and n 1 F β,A such that q ≥ q one has n+1 (1/4)−β m (ρ ) exp(−q ), ρ = 10A /(q q ). ◦f n n n+1 n F n+1 Proof. — This follows from the previous Corollary 13.5 and Proposition 12.6 and Remark 12.1. 14. Proof of Theorems C, A and A’ 14.1. Proof of Theorem C. 14.1.1. (AA) Case. — Let f (θ , r) = (θ + ω , r) + (O(r), O (r)) be a real ana- lytic symplectic diffeomorphism of the annulus T ×[−1, 1] satisfying the twist condi- tion (1.14). We can perform some steps of the classical Birkhoff Normal Form proce- dure, Proposition 6.2:for some h > 0,ρ > 0, there exists g = f = id + (O(r), O(r )), 0 Z 10h 10h 2 10h ∈ O (e D(0,ρ )),Z, F ∈ O (e (T × D(0,ρ ))) ∩ O(r ),suchthaton e (T × σ 0 σ h 0 h D(0,ρ )) one has −1 g ◦ f ◦ g = ◦ f , ∀0 ≤ ρ ≤ ρ , F 10h ≤ ρ 0 e (T ×D(0,ρ)) −1 2 3 (2π) (r) = ω r + b (f )r + O(r ) 0 2 j 10 Z(θ , r) = Z (θ )r + r Z (θ , r) j ≥10 j=2 where m is the constant appearing in (10.296). Applying Lemma 2.5 to (r) and 10 3 Lemma 2.2 to r Z (θ , r) we can ﬁnd, for some 0 < ρ ρ ,C Whitney extensions ≥10 0 10h 10h 10h h/10 ∈ O (e D(0, ρ)) and Z ∈ O (e W ) of (, e D(0, ρ)) and (Z, e W ) σ σ h,D(0,ρ) h,D(0,ρ) h/10 such that g := f ∈ Symp (e W ) (see Notations 2.3, 2.6 and 4.8), Z h,D(0,ρ) ex,σ −1 (14.393) ∈ TC(A, B), A = 3min(b (f ), b (f ) ), B ≥ 1 2 2 (14.394) g({r = 0}) = ({r = 0}), g − id 1 ≤ 1/100. Since −1 h/10 g ◦ f ◦ g = ◦ f [e W ] F h,D(0,ρ) 112 RAPHAËL KRIKORIAN one has from (4.86), for any ρ ≤ ρ , L(f , g(W )) = g(L( ◦ f , W )) D(0,ρ) F D(0,ρ) hence, using the fact that g({r = 0}) = ({r = 0}) and g − id ≤ 1/100, we get the inequality (14.395) m (ρ) m (2ρ). f ◦f TheﬁrstpartofTheorem C is then a consequence of Theorem 12.3 applied to ◦ f (which satisﬁes (10.294), (10.295)(10.296)): if we deﬁne D as the set {D, D ∈ D } (each t 2t q q D is associated to a D ∈ D ), formula (1.23) comes from the fact that #D = #D = 2t t 2t #D(U ) (recall the notation (12.370)) and from (10.309), (10.304), (10.305)); on the i (2t) other hand, (1.24) is a consequence of (12.371); ﬁnally (1.25) follows from Theorem 12.3 and inequality (14.395)(we take ρ = t). The second part of Theorem C is a consequence of Corollary 13.2 because if the BNF of f converges, the same is true for that of ◦ f . 14.1.2. (CC) Case. — Let f be a real analytic twist symplectic map of the real disk admitting the origin as an elliptic ﬁxed point with Diophantine frequency ω , (x, y) → 2 2 2 (x, y)+ O (x, y), r(x, y) = (1/2)(x + y ) and satisfying the twist condition (1.14). 2πω r(x,y) We ﬁrst make the symplectic change of variables (4.77) (z,w) = ϕ(x, y), 1 1 √ √ z = (x + iy) x = (z − iw) 2 2 ⇐⇒ i −i √ √ w = (x − iy) y = (z + iw) 2 2 −1 and we write the thus obtained symplectic map (z,w) → f (z,w), f = ϕ ◦ f ◦ ϕ as f = ◦ f , r =−izw. 2πω r F 0 0 We observe that (cf. (4.86)) (14.396) L(f , W) = L(f ,ϕ(W)). Like in the (AA)-case (Section 14.1.1) we perform some steps of Birkhoff Normal Form, Proposition 6.1 and make some Whitney extensions (Lemma 2.2)toobtainfor some h>0, 10h 10h h/10 ρ> 0, maps ∈ O (e D(0, ρ)),F ∈ O (e W ), g ∈ Symp (e W ) sat- σ σ h,D(0,ρ) h,D(0,ρ) ex,σ isfying −1 10h (14.397) g ◦ f ◦ g = ◦ f , [e W ] F h,D(0,ρ) (14.398) g({r = 0}) = ({r = 0}), g − id 1 ≤ 1/100. −1 (14.399) ∈ TC(A, B), A = 3min(b (f ), b (f ) ), B ≥ 1 2 2 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 113 (14.400) ∀ ρ ≤ ρ, F 10h ≤ ρ e W h,D(0,ρ) where m is the constant appearing in (10.296). Applying (14.396), (14.397), (14.398) yields for ρ ≤ ρ (cf. (14.395)) (14.401) m (ρ) ≤ m (2ρ) m (4ρ). f ◦f f F The conclusion of Theorem C is then obtained in the same way as in the previous Sec- tion 14.1.1. 14.2. Proof of Theorem A.— The conclusion of Theorem A is an immediate conse- quence of (1.23), (1.25), (1.26)ofTheorem C:for any β> 0and t 1 (1/τ )−β m (t) exp −(1/t) . 14.3. Proof of Theorem A’.— We proceed like in the previous Section 14.1.2 to obtain (14.397)–(14.400)and then, (14.402) m (ρ) ≤ m (2ρ) m (4ρ). f ◦f f F We now apply Corollary 13.6 to ◦ f . Setting ρ = 10A/(q q ) with A = 3min(b (f ), F n n n+1 2 −1 10 b (f ) ) (cf. (14.399)) we get for any β 1and any n 1such that q ≥ q ,the 2 β n+1 inequality (1/4)−β m (ρ ) exp(−q ). ◦f n n+1 −1 Hence if t := 5min(b (f ), b (f ) )/(q q ) ≤ ρ /4 one has (cf. (14.402)) n 2 2 n n+1 n 1/5 m (t ) ≤ m (2t ) m (4t ) exp(−q ). f n f n ◦f n F n+1 15. Creating hyperbolic periodic points Let ∈ O (D(0, ρ)) satisfy a twist condition (A, B ≥ 1), −1 2 3 (15.403) ∀ r ∈ R, A ≤ (1/2π)∂ (r) ≤ A, and (1/2π)D ≤ B, a ∈ N, a ≥ 10, be the constant appearing in Proposition G.1 of the Appendix and 3 3 −1 (p /q ) the sequence of convergents of ω = (2π) ∂(0). We introduce for n ≥ 1, the n n 0 sequence c deﬁned by −1 (2A) A −1 (15.404) (2π) ∂(c ) = p /q , ≤|c |≤ . n n n n q q q q n n+1 n n+1 114 RAPHAËL KRIKORIAN Proposition 15.1. — Let h > 0,n ∈ N large enough and F ∈ O (T × D(c ,|c | )) such σ h n n that −q h 10 e < |c | F |c | T ×D(c ,|c | ) n h n n and −q h a +1/2 n 3 |F(q , c )|≥ e |c | . n n n Then, −1 2a +1 −4q h 3 n m (c ) ≥ C |c | e . ◦f n n F h The constant C can be chosen to be non increasing w.r.t. h. This proposition will be a consequence of the more precise statement given by the following Proposition 15.2. For p ∈ Z, q ∈ N , p∧ q = 1, p/q small enough, there exists a unique c ∈ D(0, ρ)∩ p/q R such that −1 ω(c ) := (2π) ∂(c ) = p/q. p/q p/q We deﬁne −9 (15.405) ρ = min(|c /4|, q ) p/q p/q and assume that (15.406) ε := F ≤|c | . p/q D(c ,ρ ) p/q p/q p/q 2π −1 ∓iqθ The ±q-th Fourier coefﬁcients of F(·, r), F(±q, r) = (2π) F(θ , r)e dθ satisfy −qh |F(±q, r)| e ε p/q and since F is σ -symmetric, for every r ∈ D(0, ρ) ∩ R, F(q, r) = F(−q, r). Proposition 15.2. — Assume (15.406) is satisﬁed and −qh 10 (15.407) e <ρ p/q −qh (15.408) |F(±q, c )|= ν e ε . p/q q p/q −1 (15.409) ν qρ ≤ 1/q. p/q −1 −qh 3/2 Then, there exists in a neighborhood of T×{c }⊂ T× R an open set of area ≥ C (ν ε e ) , p/q q p/q C > 0, that has an empty intersection with any possible (horizontal) invariant circle of the symplectic diffeomorphism ◦ f . F ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 115 Remark 15.1. — One can choose the constant C to be non increasing with respect to h. Let us see how it provides a proof of Proposition 15.1. Proof of Proposition 15.1.— Since for n large enough −q h −9 10 e < min(|c /4|, q ) −q h 1/2 |F(q , c )|≥ e |c | F 2 n n n T ×D(c ,|c | ) h n n −1/2 −9 lim |c | q min(|c /4|, q ) = 0 n n n n→∞ 1/2 we can apply Proposition 15.2 with q = q , ε =F , ν =|c | , c = c , n p/q T ×D(c ,|c | ) q n p/q n h n n −9 ρ = min(|c /4|, q ).Wethenget p/q n 1/2 −q h 2 m (c ) (|c | e F 2 ) . ◦f n n T ×D(c ,|c | F h n n 2π −1 2 1/2 But, because ((2π) |F(θ , c )| dθ) ≥|F(q , c )|, one has F 2 ≥ n n n T ×D(c ,|c | ) n n 0 h |F(q , c )|,hence n n −1 2a +1 −4q h 3 n m (c ) ≥ C |c | e . ◦f n n The proof of Proposition 15.2 will occupy the next subsections. 15.1. Putting the system into q-resonant Normal Form. — Conditions (15.405)and (15.406) show that we can apply Proposition G.1: it provides us with the following q- resonant Normal Form −1 res g ◦ ◦ f ◦ g = ◦ ◦ f ◦ f cor F RNF 2π(p/q)r F RNF F res (15.410) = − 2π(p/q)r + M (F ) res res res F = F − M (F ) res res −1/q res res F ∈ O (T × D(c , e ρ )), F = F − M (F ); these last two functions are σ h−1/q p/q p/q 0 1/q-periodic (in the θ -variable) and are such that res −1/q F ε p/q T ×D(c ,e ρ ) h−1/q p/q p/q res res F = T (F + O(qρ F )). p/q W N h,D(c ,ρ ) p/q p/q Also, cor −1/4 −1/q (15.411) F exp(−ρ )F , e W h,D(c,ρ) h,D(c,ρ) −2 2 a −5 (15.412) g − id 1 (qρ ) F ≤ ρ . RNF C h,D(c,ρ) 116 RAPHAËL KRIKORIAN −1/q Lemma 15.3. — On T × D(c , e ρ /2) one has 1/q p/q p/q res res res ±iqθ res (15.413)F (θ , r) = u (r) + u (r)e + u (θ , r) 0 1,± ≥2 −1/q where on D(c , e ρ /2) one has p/q p/q res res u (r) = M (F ) = F(0, r) + O(qρ ε ) 0 p/q p/q (15.414) res −qh −qh u (r) = F(±q, r) + O(e qρ ε ) = O(e ε ) p/q p/q p/q 1,± and res −2qh u −1/q e ε . T ×D(c ,e ρ /2) p/q ≥2 1/q p/q p/q Proof. — We recall that from (G.526) q−res res F = T (F + G) q−res (see the notation (G.519)for T )where G −1/q = O(qρ F ). h−1/q,e ρ /2 p/q h,D(c ,ρ /2) p/q p/q p/q Hence −q(h−2/q) −qh (15.415) |G(0, r)| qρ ε , |G(±q, r)| e qρ ε e qρ ε . p/q p/q p/q p/q p/q p/q −2q(h−3/q) −2qh On the other hand since e e q−res ±iqθ −2qh T F − F(0, r) − F(±q, r)e e ε 1/q,ρ /2 p/q N p/q and q−res ±iqθ −2qh T G − G(0, r) − G(±q, r)e −1/q qρ ε e . 1/q,e ρ /2 p/q p/q N p/q Summing these two inequalities and using (15.415)gives (15.413). With these notations res = − 2π(p/q)r + u (r) res res res F = F − u (r). We denote by c ∈ R the point where ∂ (c) = 0; ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 117 res since u ε and satisﬁes the twist condition (15.403) one has D(c ,ρ ) p/q 0 p/q p/q c = c + O(ε ) ∈ D(c,(3/4)ρ ), p/q p/q 2 3 (r) = cst + ( /2)(r − c) + O((r − c) ) −1 res for some A . Since F is σ -symmetric we can write res ±iqθ u (r)e = a(r) cos(qθ) + b(r) sin(qθ) 1,± and from (15.408), (15.414), (15.409) we can assume, shifting the variable θ ∈ T by a translation θ → θ + α (α ∈ T) if necessary, that c c −qh −1 (15.416) b(c) = 0, a(c) = ν e ε , ν = ν − O(qρ ) = ν (1 + o (1)) q p/q q q p/q q q with −qh max(a ,b ) e ε . D(c,ρ ) D(c,ρ ) p/q p/q p/q Thus, res 2 3 (r) = (r) − 2π(p/q)r + u (r) = cst + ( /2)(r − c) + O((r − c) ) res res F (θ , r) = a(r) cos(qθ) + b(r) sin(qθ) + u (θ , r). ≥2 15.2. Coverings. — Like in Section 8.2 (cf. (8.215)) we deﬁne res −2/q res −2/q ∈ O (D(0, qe ρ /2)), F ∈ O (T × D(0, qe ρ /2)) σ p/q σ qh−2 p/q res 2 (15.417) (r) = q (c + r/q) res res 2 F (θ , r) = q F ([θ/q] , c + r/q) mod (2π/q)Z hence res 2 3 2 (r) = cst + r /2 + O(r ) = r /2 + ω(r) res res F (θ , r) = a(r) cos(qθ) + b(r) sin(qθ) + u (θ , r) ≥2 with 2 2 a(r) = q a(c + r/q), b(r) = q b(c + r/q), res 2 res u (θ , r) = q u (θ /q, c + r/q). ≥2 ≥2 Let us deﬁne res res res (15.418) H (θ , r) := (r) + F (θ , r) 2 res = cst + (1/2) r + a(r) cos θ + b(r) sin θ + ω(r) + u (θ , r). ≥2 118 RAPHAËL KRIKORIAN We make explicit the linear plus quadratic part H (θ , r) of (1/2) r + a(r) cos θ + b(r) sin θ at (θ , r) = (0, 0) ∈ R (recall that b(0) = 0) which appears in (1/2) r + a(r) cos θ + b(r) sin θ 2 2 2 = a(0) + r∂ a(0) − (θ /2) a(0) + (r /2)( + ∂ a(0)) + rθ∂ b(0) + g (θ , r) r 0 3 3 where g (θ , r) = O (θ , r) = O(|θ| +|r| ); we can then write 0 3 res (15.419) H (θ , r) = cst + H (θ , r) + ω(r) + g(θ , r) with θ θ 0 θ H (θ , r) = Q , + , ⎨ 2 r r ∂ a(0) r (15.420) − a(0)∂ b(0) ⎪ r Q = ∂ b(0) + ∂ a(0) and res (15.421) g(θ , r) = g (θ , r) + u (θ , r), g (θ , r) = O (θ , r). 0 0 3 ≥2 For further records we mention the following estimates 2 −qh (15.422)max( a ,b ) q e ε D(0,qρ ) D(0,qρ ) p/q p/q p/q 2 −qh (15.423) b(0) = 0, a(0) = q ν e ε , where ν = ν (1 + o (1)) q p/q q q 1/q res 2 −2qh (15.424) u −1/q q e ε T ×D(0,e qρ /2) p/q ≥2 1/q p/q and for (l , l ) ∈ N , l + l ≤ 2and 0 < t <ρ /10 1 2 1 2 p/q l l 2 3−l −l −qh 1 2 1 2 (15.425) ∂ ∂ g (θ , r) q t e ε 0 D(0,t)×D(0,t) p/q θ r l −l l 2 3−l −l −qh l 2 2 −2qh 1 2 1 2 1 (15.426) ∂ ∂ g(θ , r) [q t e ε + q ρ q e ε ]. D(0,t)×D(0,t) p/q p/q θ r p/q 15.3. Existence of a hyperbolic ﬁxed point for f .— We refer to Appendix N for the H + ω deﬁnition of the notion of a (κ, δ)-hyperbolic ﬁxed point. Lemma 15.4. — The afﬁne symplectic map f has a (κ, δ)-hyperbolic ﬁxed point H + ω(r) 5 2 2 (θ , r ) ∈ D(0,ρ ) ∩ R with 0 0 p/q −qh 1/2 δ = κ = q( ν ε e ) (1 + o (1)) p/q 1/q q ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 119 with stable and unstable directions at this point of the form where −qh 1/2 m =±q(ν e ε / ) (1 + o (1)). ± q p/q 1/q Proof. — See Appendix N.2. 15.4. Stable and unstable manifolds of f res . Lemma 15.5. — The symplectic diffeomorphism f res has a (κ, δ)-hyperbolic ﬁxed point 4 2 2 (θ , r ) ∈ D(0,ρ ) ∩ R with 1 1 p/q −qh 1/2 (15.427) κ = δ = q( ν e ε ) (1 + o (1)). q p/q 1/q The stable and unstable directions at this point are of the form where −qh 1/2 m =±q(ν e ε / ) (1 + o (1)). ± q p/q 1/q Proof. — From (15.419), (15.428) f res = f H + ω+g H Q = f ◦ f , g = O (g) H + ω g 1 and from (4.92) of Lemma 4.5,(15.426) (with l + l ≤ 2) and (N.594)weget 1 2 2 5 −qh 4 −2 −2qh Df − id [q ρ e ε + q ρ e ε ] g D(0,10θ )×D(0,10r ) p/q p/q 0 0 p/q p/q 2 5 4 −2 −qh −qh (q ρ + q ρ e )ε e . p/q p/q p/q Because of Lemma 15.4 and (15.428), the Stable Manifold Theorem N.1 of the Appendix shows that the conclusion of the Lemma is true provided for some constant C > 0(cf. (N.589)) −1 f − id 1 ≤ C ρ κδ g C (D(0,10θ )×D(0,10r )) p/q 0 0 a condition that is implied by (recall (15.427) and the fact that from (15.409) one has ν qρ ) q p/q 2 5 4 −2 −qh 2 2 −1 2 (q ρ + q ρ e )< q ρ (< C q ρ ν ). p/q q p/q p/q p/q But (15.405), (15.407) show that this last inequality is satisﬁed if q 1. 120 RAPHAËL KRIKORIAN 15.5. Stable and unstable manifolds of ◦ f . Lemma 15.6. — The diffeomorphism ◦ f has a hyperbolic q-periodic point (θ, r) the local stable and unstable manifolds of which are graphs of C -functions w ,w :]θ − ρ, θ + ρ[→ R such − + that (m/2)|θ − θ|≤|w (θ ) − w (θ )|≤ 2m|θ − θ| (for θ ∈]θ − ρ, θ + ρ[) + − −qh 1/2 m = q(ν e ε / ) (1 + o (1)) q p/q 1/q −1 −qh ρ = C ν e ε . q p/q Proof. — Recall, cf. (15.410), that −1 res (15.429) g ◦ ◦ f ◦ g = ◦ ◦ f ◦ f cor . F RNF 2π(p/q)r F RNF F From (15.417), the pre-image of (θ , r ),by (θ , r) → ([(θ − α )/q] , c + r/q) is 1 1 c mod (2π/q)Z 4 4 a q-periodic orbit O ⊂ T×]c − ρ , c + ρ [⊂ T×]c − ρ /3, c + ρ /3[ of q p/q p/q p/q p/q p/q p/q p/q p/q res res res ◦ f as well as of ◦ ◦ f (F is 2π/q-periodic); Lemma 15.5 tells us that 2π(p/q)r F F 4 4 this periodic orbit is hyperbolic. Let u ∈ T×]c − ρ , c + ρ [ be a point of O and 0 p/q p/q q p/q p/q res denote ϕ = ◦ ◦ f . One has ϕ (u ) = u and we want to ﬁnd a hyperbolic 2π(p/q)r 0 0 cor cor ﬁxed point for (ϕ ◦ f ) (the q-th iterate of ϕ ◦ f )close to u . F F 0 We can write q q (ϕ ◦ f cor ) = ϕ ◦ j where −(q−1) q−1 −1 cor cor cor j = (ϕ ◦ f ◦ ϕ )◦···◦ (ϕ ◦ f ◦ ϕ) ◦ f . F F F Since ( ◦ ) 2 1 uniformly in n and 2π(p/q)r C (T×]c −ρ /3,c +ρ /3[) p/q p/q p/q p/q res −2 F 2 ε ρ 1 C (T×]c −ρ /3,c +ρ /3[) p/q p/q p/q p/q p/q p/q one has for n ≤ ρ /ε , p/q p/q (15.430) ϕ 1 C (T×]c −ρ /3,c +ρ /3[) p/q p/q p/q p/q −1 and consequently (q ε ) cor j − id 1 qF 1 C (T×]c −ρ /3,c +ρ /3[) C (T×]c −ρ /3,c +ρ /3[) p/q p/q p/q p/q p/q p/q p/q p/q −1/3 exp(−ρ ) p/q where we have used (15.411). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 121 q q −1 −1 Replacing ϕ and j by T ◦ ϕ ◦ T and T ◦ j ◦ T where T : u → u − u we can 4 2 2 assume that u = 0 ∈ D ( c,ρ ) ⊂ D ( c,ρ /3) ⊂ T×] c −|c /3|, c + ρ /3[.We 0 R R p/q p/q p/q p/q q q then have ϕ (0) = 0and the matrixDϕ (0) is (κ, δ)-hyperbolic with 2 −qh δκ = q ν e ε (1 + o (1)). q p/q 1/q q q Write ϕ (u) = Dϕ (0)ξ (u) with ξ(0) = 0, Dξ(0) = id so that q q ϕ ◦ j = Dϕ (0) ◦ ξ ◦ j. Observe that for 0 <ρ <ρ /4and k = 0, 1 p/q k k D (ξ ◦ j − id) 0 2 D (ξ − id) 0 2 +j − id 1 2 C (D (0,ρ)) C (D (0,ρ)) C (D (0,ρ)) R R R −1/3 2−k ρ + q exp(−ρ ). p/q Let us choose −1 −qh ρ = C ν e ε q p/q with C large enough. The Stable Manifold Theorem (cf. Appendix, Theorem N.1)shows that the diffeomorphism ϕ ◦ j has a hyperbolic ﬁxed point the stable and unstable man- ifolds of which are graphs of C functions of the form w , w :] − ρ/2,ρ/2[→ R, − + w < 0 < w ,suchthatfor all θ ∈] − ρ/2,ρ/2[ one has − + (3/2)m θ ≤ w (θ ) ≤ (2/3)m θ ≤ 0 ≤ (2/3)m θ< w (θ ) ≤ (3/2)m θ. − − − + + + −1 To conclude the proof of the Lemma we set w = w ◦ g and note that ± ± RNF −1 q g ◦ ◦ f ◦ g = ϕ ◦ j F RNF RNF −1 with g − id ≤ 1/10 (see (15.412)). RNF 15.6. End of the proof of Proposition 15.2.— Let V be the set V={(θ , r), θ ∈[θ, θ + ρ/2],w (θ ) ≤ r ≤ w (θ )} − + the boundary of which is made by two pieces of stable and unstable manifolds and the vertical segment L := {θ + ρ/2}×[w (θ + ρ/2), w (θ + ρ/2]. By a theorem of Birkhoff − + [4](cf. also [21]), any invariant (horizontal) curve of the twist diffeomorphism ◦ f is the graph of a Lipschitz function γ : T →[−1, 1]; if this curve intersects the stable or unstable manifold of (θ, r) it must be included in the union of these stable and unstable manifolds which is impossible. So if this invariant curve intersects the interior of V it has to enter in V by ﬁrst entering the vertical segment L by the right. But this is clearly impossible also (see Figure 11). 122 RAPHAËL KRIKORIAN γ(·) w (·) w (·) (θ, r) V V θ + ρ/2 FIG. 11. — Invariant graphs cannot intersect the interior of V Now the domain V has an area which is area(V) ρ × (m − m ) + − −qh 3/2 (ν e ε ) . q p/q This concludes the proof of Proposition 15.2 if we notice that the dependence on h of the implicit constant in the symbol appears only when we apply Proposition G.1 on Resonant Normal Forms (cf. Remark G.1). 16. Divergent BNF: proof of Theorems E, B and B’ We now use the result of the previous Section to construct examples of real analytic symplectic diffeomorphisms of the disk and the annulus with divergent BNF. 16.1. Proof of Theorems E and B: the (AA) Case. — Let f = ◦ f be a real 2πω r O(r ) analytic symplectic twist map of the annulus of the form (1.6) and satisfying the twist condition (1.14). We perform a Birkhoff Normal Form, cf. Proposition 6.2,on f up to order a ,where a is the integer of Proposition G.1 of the Appendix that appears in 3 3 Proposition 15.1: there exist ρ> 0, g ∈ Symp (T × D(0, ρ)) exact symplectic, ∈ ex,σ O (D(0, ρ)),F ∈ O (T × D(0, ρ)) such that σ σ h −1 g ◦ f ◦ g = ◦ f where (b = 0) −1 2 3 a 2 (2π) (r) = ω r + b r + O(r ), F(θ , r) = O(r ), g − id = O(r ). 0 2 −1 Note that for ρ small enough satisﬁes a ((5/2) min(b , b ), B)-twist condition on D(0, ρ) (B ≥ 1). In particular, if (p /q ) are the convergents of ω and c ∈ R (n large n n n≥1 0 n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 123 enough) is the point where −1 (5/2) max(b (f ), b (f ) ) 2 2 −1 (16.431) (2π) ∂(c ) = p /q , |c |≤ n n n n q q n n+1 (cf. (15.404) and the twist condition satisﬁed by ) one has F 2 |c | . T ×D(c ,|c | ) n h n n For (ζ ) ,(ζ ) ∈[−1, 1] ,let G ∈ T × D(0, 1) deﬁned by 1,k k≥1 2,k k≥1 ζ h a −q h −q h 3 k k G (θ , r) = r ζ e cos(q θ) + ζ e sin(q θ). ζ 1,k k 2,k k k≥1 We now deﬁne f ∈ Symp (T × D(0, 1)),F ∈ O (T × D(0, ρ)) by ζ ζ σ h f = f ◦ f , ζ G −1 −1 ◦ f := g ◦ f ◦ g = ◦ f ◦ g ◦ f ◦ g. F ζ F G ζ ζ Lemma 16.1. — For n, c as above, there exists a set J (F) ⊂[−1, 1] of 2-dimensional n n 1/2 2 N 2 Lebesgue measure |c | such that for any ζ ∈ ([−1, 1] ) , such that ζ ∈[−1, 1] J (F) one n n n has 2a +1 −4q h 3 n m (c ) |c | e . ◦f n n Proof. — Let α ∈ T and ν ≥ 0be such that n q iq θ −iq θ a −q h n n 3 n F(q , c )e + F(−q , c )e =|c | ν e cos(q θ + α ). n n n n n q n q n n −1 a 2 Since ◦ f = ◦ f ◦ g ◦ f ◦ g,F, G = O(r ) and g − id = O(r ) we see that F F G ζ ζ ζ a +1 (16.432)F = F + G + O(r ). ζ ζ We now assume c > 0 for simpler notations (the case c < 0 istreated in thesameway). n n We can write a −q h −q h 3 k k G (θ , r) = r ζ e cos(q θ + α ) + ζ e sin(q θ + α ) ζ 1,k k q 2,k k q k k k≥1 −iα with ζ − iζ = e (ζ − iζ ) and from (16.432) we see that 1,k 2,k 1,k 2,k iq θ −iq θ n n F (q , c )e + F (−q , c )e ζ n n ζ n n a −q h 3 n =|c | e (ν +ζ +u (ζ )) cos(q θ+α )+(ζ +v (ζ )) sin(q θ+α ) n q 1,n n n q 2,n n n q n n n 124 RAPHAËL KRIKORIAN where sup (|u (ζ )|,|v (ζ )|) |c |. n n n 2 N ζ∈([−1,1] ) We can thus write for ζ ,ζ ∈] − 1, 1[ 1 2 a −q h 3 n 2|F (q , c )|= ν (ζ )|c | e ζ n n n n with 2 2 2 ν (ζ ) = (ν + ζ + u (ζ )) + (ζ + v (ζ )) n q 1,n n 2,n n ≥|ζ − O(c )| . 2,n n −iα 2 Since ζ − iζ = e (ζ − iζ ), one can hence ﬁnd a set J (F) ⊂[−1, 1] of 2- 1,k 2,k 1,k 2,k n dimensional Lebesgue measure 1/2 |J (F)| |c | n n such that 2 1/2 (ζ ,ζ )∈[−1, 1] J (F) %⇒ |ν (ζ )| |c | . 1,n 2,n n n n By Proposition 15.1 we thus have 2a +1 −4q h 3 n m (c ) |c | e . ◦f n n 2 N Lemma 16.2. — Let N ⊂ N be inﬁnite. Then, for almost every ζ ∈ ([−1, 1] ) for the ⊗N product measure μ = (Leb 2 ) , there exists an inﬁnite subset N ⊂ N such that for all n ∈ N ∞ [−1,1] 2a +1 −4q h 3 n m (c ) |c | e . ◦f n n Proof. — Since the random variables ζ , n ∈ N are independent, for any m ∈ N, the event {ζ ∈ J (F), ∀ n ≥ m} has zero μ -probability as well as their union. Hence for n n ∞ μ -almost every ζ ∈ X , one has for inﬁnitely many n ∈ N , ζ ∈ / J (F) and we conclude ∞ n n by Lemma (16.1). 16.1.1. Proof of Theorems E and B.— We now observe that if ω is Diophantine with exponent τ ln q n+1 τ = lim sup , ln q then for any β> 0 there exists a inﬁnite set N such that for all n ∈ N β β τ−β/4 q ≥ q . n+1 n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 125 On the other hand p 1 1 |c | ω − n 0 1+τ−β/4 q q q n n n+1 n hence (1/(1+τ))+β/4 q (1/|c |) n n and consequently, from Lemma 16.2, for an inﬁnite number of n ∈ N ( )+β/2 1+τ 2a +1 −4q h (16.433) m (c ) |c | e exp − . ◦f n n |c | We observe that since t ≥ 2|c | (cf. (1.20)and (16.431)) one has n n ( )+β 1+τ (16.434) m (t ) m (c ) exp − . f n ◦f n ζ F |t | If β is chosen so that 1 1 + β< − β, 1 + τ τ the estimate (16.434), when compared to the conclusion of Theorem A, shows that the 2 N Birkhoff Normal Form of f is divergent for μ -almost every ζ ∈ ([−1, 1] ) . ζ ∞ This concludes the proof of Theorem E and, as a consequence, of that of Theo- rem B (in the (AA) case). 16.2. Proof of Theorems E’, B and B’: (CC) Case. — Let f be a real analytic symplectic diffeomorphism of the disk admitting the origin 0 as an elliptic equilibrium with irrational frequency ω and satisfying the twist condition (1.14); we assume that it is of the form 2 2 a 2 2 f = + O((x + y ) ) ((1/2)(x +y )) with ∈ O (D(0, 1)). Passing to the (z,w)-variables (cf. (4.77)) we can write −1 ϕ ◦ f ◦ ϕ = ◦ f where F ∈ O (D(0, 1) ) a a 3 3 F(z,w) = O((zw) )(and not only O (z, w)). Let as before (p /q ) be the convergents of ω and c ∈ R the point where n n n≥1 0 n −1 (2π) ∂(c ) = p /q (cf. (15.404)). n n n 126 RAPHAËL KRIKORIAN 2 N 2 For (ζ ) ∗ ∈ ([−1, 1] ) ,let G ∈ O (D(0, 1) ) n n∈N ζ σ 1,k a −1/2 q −1/2 q 3 k k G (z,w) = (−izw) × ((i z) + (i w) ) k=1 2,k −1/2 q −1/2 q k k + × ((i z) − (i w) ). 2i We now deﬁne (recall the deﬁnition of G in Section 1.4.3) ◦ f = ◦ f ◦ F F G ζ ζ −1 f = ϕ ◦ ( ◦ f ) ◦ ϕ = f ◦ = f ◦ .(cf .(4.82)) ζ F G ◦ϕ G ζ ζ Lemma 16.3. — Assume that for some n large enough, c is positive. Then, there exists J (F) ⊂ n n 2 1/2 [−1, 1] of Lebesgue measure c such that if ζ = (ζ ,ζ )/∈ J (F) one has n 1,n 2,n n 2a +1 −4q h 3 n m (2c ) m (c ) c e . f n ◦f n ζ F Proof. — We deﬁne h =−(1/2) ln(c + c ), n n n 1 and since |ω − | c one has 0 n q q q n n n+1 h = (−1/2) ln c + O(c ) n n n = (−1/2) ln c − O(1/q ) hence −q h q /2 O(1/q ) 10 n n n n (16.435) e = c e < c . n n CC CC 2 h 2 1/2 Let W = W ={(z,w) ∈ C , max(|z|,|w|) ≤ e (c + c ) , −izw ∈ 2 n n n h ,D(c ,c ) n n D(c , c )}. One has a a 3 3 CC CC (16.436) F |c | , G c . W n ζ W n n n Using Lemma K.1 we can pass to (AA)-coordinates: if ψ is the diffeomorphism deﬁned in (4.79) −1 CC AA 2 ψ (W ) ⊃ W = T × D(c , c ) 2 2 h n − h ,D(c ,c ) h ,D(c ,c ) n n n n n n n AA AA 2 and we can introduce F , F ∈ O (T × D(c , c )) (cf. (4.82)) σ h n ζ n −1 −1 ψ ◦ f ◦ ψ = f AA,ψ ◦ f ◦ ψ = f AA = f AA ◦ . F − F F − F F G ◦ψ − − ζ ζ − ζ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 127 AA AA AA AA Since F = F ◦ ψ + O (F) and F = F + G ◦ ψ + O (F , G ◦ ψ ) (cf. (4.94), − 2 ζ − 2 ζ − AA (4.92)) one has on W AA a AA AA (3/2)a 3 3 (16.437) F AA c , F = F + G ◦ ψ + O(c ). W ζ − n n ζ n ζ If we deﬁne ν and α ∈ T by n n AA iq θ AA −iq θ a −q h n n 3 n n F (q , c )e + F (−q , c )e =|c | ν e cos(q θ + α ) n n n n n q n q n n we see that on T × D(c , c /2) (cf. (16.437)) h −1 n n n AA AA a q /2 (3/2)a 3 k 3 F = F (θ , r) + r r (ζ cos(q θ) + ζ sin(q θ)) + O(c ). 1,k k 2,k k ζ n k=1 Hence AA iq θ −iq θ n n F (q , c )e + F (−q , c )e n n ζ n n a (1/2)a −q h q /2 3 3 n n n = c (ν + O(c ))e + c ζ cos(q θ + α ) q 1,n n q n n n n n q /2 (1/2)a −q h n 3 n n + c ζ + O(c )e sin(q θ + α ) 2,n n q n n −iα with ζ − iζ = e (ζ − iζ ).Wethushave(cf. (16.435)) 1,k 2,k 1,k 2,k AA a q /2 (1/2)a −q h 3 n 3 n n 2|F (q , c )|≥ c |c ζ + O(c )e | n n 2,n ζ n n n a −q h O(1/q ) (1/2)a −q h 3 n n n 3 n n ≥ c |e e ζ + O(c )e | 2,n n n a −q h (1/2)a 3 n n 3 c e |ζ + O(c )| 2,n n n and we see that if 1/2 |ζ |≥ c 2,n one can apply Proposition 15.1 (cf. (16.435)): −1 2a +1 −4q h 2a +1 −4q h 3 n 3 n m (c ) C c e c e . ◦f n AA h n n Now, since c is positive and m (2c ) m (c ) this provides n f n ◦F n ζ ζ 2a +1 −4q h 3 n (2c ) m (c ) c e . f n ◦f ◦ n ζ F G We can deduce the analogue of Lemma 16.2 128 RAPHAËL KRIKORIAN Lemma 16.4. — Let N be an inﬁnite set of n ∈ N for which c > 0. Then, for almost every 2 N ζ ∈ ([−1, 1] ) , there exists an inﬁnite subset N ⊂ N such that for all n ∈ N 2a +1 −4q h 3 n m (2c ) m (c ) c e . f n ◦f n ζ F 16.2.1. Proof of Theorems E’ and B (CC) Case, ω Diophantine. — We want to apply thepreviousLemma 16.4 to an inﬁnite set N such that for all n ∈ N one has both τ− (16.438) c >0and q ≥ q . n n+1 Such a set may not exist for arbitrary choices of ω (Diophantine) and . On the other hand, if one chooses the sign of ∂ (0) depending on ω (or more precisely its sequence of convergents) this is possible. Let β> 0and τ−β/2 N ={n ∈ N, q ≥ q }, Q ={p /q , n ∈ N }. β n+1 β n n β Since N is inﬁnite, one of the two sets Q = Q ∩ (]ω ,±∞[) is inﬁnite. We deﬁne β β 0 s (ω ) as the non-empty subset of {−1, 1} such that ±1 ∈ s (ω ) if and only if Q is β 0 β 0 inﬁnite. −1 We now assume that ω(r) := (2π) ∂(r) is of the form ω(r) = ω + 2b r + O(r ), with sign(b ) ∈ s (ω ). 0 2 2 β 0 For the sake of simplicity we shall assume that 1 ∈ s(ω ) and b > 0(the case −1 ∈ s(ω ) 0 2 0 and b < 0 is treated similarly). The sets + −1 + N := N ={n ∈ N , p /q >ω }, C = ω (Q ) β n n 0 β β β β are then inﬁnite; note that C ⊂]0,∞[ and its points c , n ∈ N accumulate zero. β n We then apply Lemma 16.4: for almost every ζ and inﬁnitely many n ∈ N 2a +1 −4q h 3 n m (2c ) c e f n ζ n and, arguing like in Subsection 16.1.1, we see that setting t = 2c we have for inﬁnitely n n many n ∈ N = N ( )+β 1+τ m (t ) exp − . f n |t | 16.2.2. Proof of Theorems E’ and B’: (CC) Case, ω Liouvillian. — Since ω is Liouvil- 0 0 lian, there exists an inﬁnite set N ⊂ N such that ln q n+1 lim =∞. n∈N ln q n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 129 We deﬁne Q ={p /q , n ∈ N , p /q ∈]ω ,±∞[} n n n n 0 + − and s(ω ) as the non-empty subset of {−1, 1} (one of the two sets Q , Q is inﬁnite) such that ±1 ∈ s(ω ) if and only if Q is inﬁnite. We assume that 1 ∈ s(ω ) and b > 0(the case −1 ∈ s(ω ) and b < 0is treated 0 2 0 2 + −1 + similarly) and we set N ={n ∈ N , p /q >ω }. The set C = ω (Q ) is inﬁnite, con- n n 0 tained in ]0,∞[ and its points c , n ∈ N accumulate 0. We now apply Lemma 16.4: for almost every ζ one has for inﬁnitely many n ∈ N 2a +1 −4q h 3 n m (2c ) c e . f n ζ n For these n’s one has p 1 c |ω − | , n 0 q q q n n n+1 and, for any β> 0, provided n is large enough, β/2 β 2a +1 −4q h 2(2a +1) 3 n 3 c e ( ) exp(−q ) exp(−q ). n+1 n+1 n+1 If we set t = 2c (cf. (16.431)) n n −1 5(b + b ) 2c ≤ t := n n q q n n+1 hence m (t ) ≥ m (2c ) exp(−q ) f n f n ζ ζ n+1 for inﬁnitely many n in N := N . Acknowledgements The author wishes to thank Alain Chenciner and Håkan Eliasson for their con- tinuous encouragements, Abed Bounemoura, Bassam Fayad, Jacques Féjoz, Jean-Pierre Marco, Stefano Marmi, Laurent Niederman, Ricardo Pérez-Marco, Laurent Stolovitch for interesting discussions and all the participants of the Groupe de travail de Jussieu for their patient listening of this work and their constructive comments. The author is also very grateful to the referees the comments of which helped improving signiﬁcantly this text. He is in particular indebted to one of the referees who pointed out an error in the exponent of Theorems A and C in the submitted version of this paper. 130 RAPHAËL KRIKORIAN Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. 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Appendix A: Estimates on composition and inversion A.1 Proof of Lemma 4.4.— We shall do the proof in the (AA)-case; the proof in the (CC)-case follows the same lines. −δ We can assume that e W = W −δ is not empty (otherwise, there is nothing h,U h−δ/2,e U to prove). By (2.53), there exists a numerical constant C > 0such that if δ> 0satisﬁes −2 −1 (A.439)CF δ d(W ) < 1, W h,U h,U −δ then for any ﬁxed θ ∈ T and any ﬁxed r ∈ e U, the map U → U, R → r − ∂ F(θ , R) h−2δ θ is contracting and by the Contraction Mapping Principle there thus exists a unique R ∈ U −δ depending holomorphically on (θ , r) ∈ T × e Usuch that h−δ/2 r = R + ∂ F(θ , R). On the other hand, assumption (A.439) and Cauchy’s inequality (2.53) show that if C is large enough −1 F(θ , R)| δ ×F <(1/2)δ, |∂ R h,U hence ϕ := θ + ∂ F(θ , R) ∈ T . R h−δ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 131 We can thus deﬁne a holomorphic map −δ f : T × e U → T × U F h−δ/2 h−δ by r = R + ∂ F(θ , R) (A.440) f (θ , r) = (ϕ, R) ⇐⇒ ϕ = θ + ∂ F(θ , R). Notice that the maps (θ , r) → ϕ(θ, r)− θ , (θ , r) → R(θ , r)− r such deﬁned are Lipschitz −2 −2 with Lipschitz constant δ d(U) F . Thus, if for some numerical constant large h,U enough −2 −2 (A.441)Cδ d(U) F < 1, h,U −δ the map f is a holomorphic diffeomorphism from T × e U onto its image. F h−δ/2 −2δ Conversely, if (A.439)issatisﬁed, given (ϕ, R) ∈ T × e U, thesamearguments h−δ −δ as those developed above show there exists a unique (θ , r) ∈ T × e Usuch that h−δ/2 f (θ , r) = (ϕ, R).Wethushaveif(A.441)issatisﬁed −2δ −δ T × e U ⊂ f (T × e U). h−δ F h−δ/2 Finally, we observe that the diffeomorphism f is exact symplectic which means that the differential form Rdϕ − rdθ is exact; in particular, it is symplectic. Indeed Rdϕ − rdθ =−ϕd R + d(ϕR) − rdθ =−(θ + ∂ F(θ , R))d R − (R + ∂ F(θ , R))dθ + d(ϕR) R θ =−d F + d(ϕR) − d(θ R) = d(−F + (ϕ − θ)R) (observe that the function −F + (ϕ − θ)R =−F(θ , R) + ∂ F(θ , R)R is well deﬁned on T × U). We have thus proven that there exists a numerical constant C > 0such that if −2 −2 (A.442) Cδ d(U) F < 1 h,U the diffeomorphism f previously deﬁned is exact symplectic and −2δ −δ (A.443) e W ⊂ f (e W ) ⊂ W . h,U F h,U h,U Estimate (4.91) comes from (A.440)and max|∂ F(θ , R) − ∂ F(ϕ, r)|| ≤ 2D FDF. i i i=1,2 132 RAPHAËL KRIKORIAN A.2 Proof of Lemma 4.5.— We illustrate the proof in the (AA)-Case (it is the same in the (CC)-case). Since f is close to the identity, the map f : (θ , R) → (ϕ, r) ⇐⇒ f (θ , r) = (ϕ, R) deﬁnes a diffeomorphism such that f − id = O(f − id) and since f is exact symplectic we know (cf. Section A.1)that ϕd R + rdθ = d F for some holomorphic function F : (θ , R) → F(θ , R). Since F(θ , R) = (ϕd R + rdθ) where γ is a path joining (0, R )∈{θ ∈ C, θ,R 0 θ,R |$θ| < h}× Uto (θ , R), the function F, which is unique up to the addition of a constant, thus satisﬁes F = O(f − id). The estimate (4.92) is a consequence of (4.87) and the fact that |∂ F(θ , R) − ∂ F(θ , r)|≤D∂ F|R − r|≤D∂ F∂ F θ θ θ θ θ |∂ F(θ , R) − ∂ F(θ , r)|≤D∂ F|ϕ − θ|≤D∂ F∂ F. R R R R R A.3 Proof of Lemma 4.6. — 1) Proof of (4.95). One has r = R + ∂ F(θ , R) f (θ , r) = (ϕ, R) ⇐⇒ ϕ = θ + ∂ F(θ , R) R = Q f (ϕ, R) = (ψ, Q) ⇐⇒ ψ = ϕ + ∂ (Q) hence Q = Rand ψ = ϕ + ∂ (R) = θ + ∂ F(θ , R) + ∂ (R) R R = θ + ∂ ( + F)(θ , Q) thus, since r = R+ ∂ F(θ , R) = Q+ ∂ F(θ , Q) and does not depend on the θ -variable, θ θ one has r = Q + ∂ ( + F)(θ , Q) ψ = θ + ∂ ( + F)(θ , Q) which is equivalent to f (θ , r) = (ψ, Q) = f ◦ f (θ , r). +F F 2) Proof of (4.93). Assume that f (θ , r) = (ϕ, R) and f (ϕ, R) = (ψ, Q).Then F G r = R + ∂ F(θ , R) (A.444) f (θ , r) = (ϕ, R), ϕ = θ + ∂ F(θ , R) R ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 133 R = Q + ∂ G(ϕ, Q) (A.445) f (ϕ, R) = (ψ, Q), ψ = ϕ + ∂ G(ϕ, Q) Qdψ − rdθ = Qdψ − Rdϕ + Rdϕ − rdθ = d(−F − G + (ϕ − θ)R + (ψ − ϕ)Q). If f ◦ f = f then one has Qdψ − rdθ = d(−H + Q(ψ − θ)) and then G F H 0 = d(−H + F + G + Q(ψ − θ) − R(ϕ − θ) − Q(ψ − ϕ)) = d(−H + F + G − (Q − R)(ϕ − θ)) and so H(θ , Q) = cst + F(θ , R) + G(ϕ, Q) − (Q − R)(ϕ − θ). Let us write H(θ , Q) = F(θ , Q) + G(θ , Q) + A(θ , Q) where −A = F(θ , Q) − F(θ , R) + G(θ , Q) − G(ϕ, Q) + (Q − R)(ϕ − θ) = F(θ , Q) − F(θ , R) + G(θ , Q) − G(ϕ, Q) − ∂ G(ϕ, Q)∂ F(θ , R) ϕ R We can now estimate A ≤∂ F Q − R +∂ G ϕ − θ h−δ,U R h,U h,U ϕ h,U h,U +∂ G(ϕ, Q) ∂ F(θ , R) ϕ h,U R h,U ≤∂ F ∂ G +∂ G ∂ F R h,U ϕ h,U ϕ h,U R +∂ G ∂ F ϕ h,U R h,U and deduce (4.93). 3) Proof of (4.94). We just write −1 f ◦ f = f ◦ f 2 F+G F+G −G+O(|D G|DG|) = f (using (4.93)) F+DFO (G) −1 and a similar expression for f ◦ f = f 2 . F+G −F+O(|D F||DF|) The proof of (4.93)and (4.94) is the same in the (CC)-case. A.4 Proof of Proposition 4.7.— We ﬁrst state two lemmata. Lemma A.1. — Let W be an open subset of M = C or T × C, v ∈ O(W) and g − id ∈ O(W) such that g − id 1. Then if v is small enough W W −1 (A.446) (id + v) ◦ g ◦ (id + v) = g ◦ (id +[g]· v + O (v)) 2 134 RAPHAËL KRIKORIAN where −1 (A.447) [g]· v =−v + (Dg · v) ◦ g. Proof. — One has −1 (id + v) ◦ g ◦ (id + v) = g ◦ (id − v + O (v)) + v ◦ g ◦ (id − v + O (v)) 2 2 = g − Dg · v + v ◦ g + O (v) hence −1 −1 g ◦ (id + v) ◦ g ◦ (id + v) −1 = g ◦ g − Dg · v + v ◦ g + O (v) −1 −1 = id − Dg ◦ g · Dg · v + Dg ◦ g · v ◦ g + O (v) −1 = id − v + (Dg · v) ◦ g + O (v). Lemma A.2. — If ∈ O(U), Y ∈ O(W ∪ (W ))) then h,U h,U −1 (A.448) f ◦ ◦ f = ◦ f Y []·Y+O (Y) where []· Y = Y ◦ − Y. Proof. — From Lemma A.1,and (4.92)wehave −1 (A.449) f ◦ ◦ f = ◦ (id +[ ]· (J∇ Y) + O (Y)) Y 2 −1 = ◦ id − J∇ Y + (D · (J∇ Y)) ◦ + O (Y) . On the other hand (A.450)J∇(Y ◦ ) = J ( D ) · (∇ Y ◦ ). t −1 Because is symplectic, one has J (D ) = (D ) J, and we deduce from both (A.449), (A.450)that −1 f ◦ ◦ f = ◦ (id + J∇ Y − J∇(Y ◦ ) + O (Y)) Y 2 = ◦ f []·Y+O (Y) where []· Y = Y ◦ − Y. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 135 From (4.92), (4.93)(we usethe fact that D(O(DFG))=DFO (G)) −1 (A.451) f ◦ f ◦ f = f Y F F+DFO (Y) Y 1 and on the other hand, from Lemma A.2 −1 (A.452) f ◦ ◦ f = ◦ f . Y []·Y+O (Y) Y 2 Hence (we use (4.93) in the last line of the following equations) −1 −1 −1 (A.453) f ◦ ◦ f ◦ f = f ◦ ◦ f ◦ f ◦ f ◦ f Y F Y Y F Y Y Y (A.454) = ◦ f ◦ f []·Y+O (Y) F+DFO (Y) 2 1 (A.455) = ◦ f , F+[]·Y+DFO (Y) and the conclusion follows if F = F+[]· Y+DFO (Y). Appendix B: Whitney type extensions B.1 Proof of Lemma 2.2.— Let χ : R→[0, 1] be a smooth function with support −δ/2 −δ/2 in [−1, 1] and equal to 1 on [−e , e ] such that j −j (B.456)sup|∂ χ | δ . δ/2 2 2 −δ We deﬁne for r ∈ C, η(r) = χ ((e |r|) /ρ ) and for i ∈ J , η (r) = (1 − χ ((e |r − δ U i δ 2 2 −δ −δ/2 c |) /ρ )).Notethat η is equal to 1 on e D(0,ρ) and 0 on C e D(0,ρ) and η is i i δ δ/2 equal to 1 on C e D(c ,ρ ) and 0 on e D(c ,ρ ) hence ζ = η η is equal to 1 on i i i i i i∈J −δ −δ/2 δ/2 e Uand 0 onV := (C e D(0,ρ)) ∪ e D(c ,ρ ). The union of the open sets i i i∈J −1/10 W −δ/10 (resp. e U) and W (resp. V) is W (resp. C) and on their intersection the h,e U h,V h,C functions ζ F and 0 coincide. As a consequence, one can extend ζFby 0 on W as a h,V Wh smooth function F : W → W . Note that since ζ is σ -symmetric, the same is true h,C h,C Wh Wh −δ for F and that F and F coincide on W −δ (which contains e W ). h,e U h,U Wh To get the estimates on the derivatives of F we observe from (B.456)and the deﬁnitions of η, η that j j −k −2k −2k max(max sup|D η|, max sup|D η |) δ max(ρ ,ρ ) i 0≤j≤k 0≤j≤k i C C and since max (η, η ) ≤ 1, one has by Leibniz formula i i j k −k −2k −2k max sup|D ζ| (#J + 1) δ max(ρ ,ρ ). 0≤j≤k i Wh Hence F := ζ Fsatisﬁes j Wh k −2k j sup D F ≤ C(1 + #J ) (δ d(U)) maxD F . W U W h,C −δ/10 h,e U 0≤j≤k 0≤j≤k 136 RAPHAËL KRIKORIAN −n −1 n B.2 Proof of Lemma 2.5.— Write (2π) (z) = b z with |b |≤ ρ , n n n=0 0 −1 2 −1 n (2π) (z) = b + b z + b z , (2π) (z) = b z .For 0 ≤ j ≤ 3and δ> 0, 2 0 1 2 ≥3 n n=3 there exists C > 0such that for any ρ ≤ ρ /2 j 0 j 3−j (B.457) D ≤ C ρ . ≥3 D(0,ρ) j Let χ : C →[0, 1] be a real symmetric smooth function with support in D(0, 1) and equal to 1 on D(0, 1/2). We deﬁne the real symmetric function deﬁned on C Wh (z) = (z) + χ(z/ρ) (z). 2 ≥3 Wh For any z ∈ D(0,ρ/2) one has (z) = (z) and by (B.457) and Leibniz formula, for j j some constant B depending only on b , b , b , D χ 0 , D ,0 ≤ j ≤ 3, one 0 1 2 C ≥3 D(0,ρ ) has 3 Wh ∀z ∈ C, D (z) ≤ B. 2π j j On the other hand, for some constant C depending only on, ∂ χ 0 , ∂ ,0 ≤ C D(0,ρ ) j ≤ 2 2 Wh ∀ t ∈ R, ∂ (t) − 2b ≤ Cρ 2π and if ρ = ρ is chosen small enough so that Cρ< b , one has (we assume b > 0) b ≤ 2 2 2 2 Wh ∂ (t) ≤ 3b . 2π B.3 Proof of Proposition 2.7.— The proof will follow from the following two lemmas. Lemma B.1. — Let β ∈ R, ν> 0;iffor some t + is ∈ U (t, s ∈ R)one has |ω(t + is) − β| <ν, then |ω(t) − β|≤ (7/6)ν (B.458) |s|≤ (4/3)Aν. Proof. — Since ω is holomorphic on U one has for any z ∈ U, ∂ω(z) = 0(we use in this proof the usual notations ∂ = (1/2)(∂ + i∂ ) and ∂ = (1/2)(∂ − i∂ )). For any point t s t s z ∈ D(0,ρ) one has (cf. Lemma 2.1) dist(z, U) ≤ 2a(U) ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 137 and from the fact that D∂ω≤ B we thus get using condition (2.60) (B.459) ∂ω 0 ≤ a(U)B C (D(0,ρ)) −1 ≤ (8A) . Now, we write (B.460) ω(t + is) − β = ω(t) − β + ∂ω(t) · (is) + ∂ω(t) · (−is) + O(s ) where 2 2 2 (B.461) |O(s )|≤D ω 0 × s C (D(0,ρ)) ≤ B × ρ × s −1 ≤ (8A) × s (cf. (2.60)). Note that since ω is real-symmetric, ∂ω(r) and ∂ω(r) are real when r is real. Hence if |ω(t + is) − β| <ν for some t + is ∈ U, one gets by taking the imaginary part in −1 (B.460), using (B.459), ∂ω(t)∈[A , A] and (B.461), that (B.462) |s|≤ (4/3)Aν. This, (B.461) and taking the real part of (B.460) show that (B.463) |ω(t) − β|≤|#(ω (t + is) − β)|+|#(O(s ))|≤ (7/6)ν. −1 Because t → ω(t) is increasing with a derivative bounded below by A (this is the twist condition) the set of t ∈] − ρ, ρ[:= D(0,ρ) ∩ R such that |ω(t) − β|≤ (7/6)ν is a (possibly empty) interval I of length ≤ (7/3)Aν . 2 −1 Lemma B.2. — Let ν ∈]0,(6A B) [.If I is not empty there exists a unique c ∈] − ρ − β β 2Aν, ρ + 2Aν[ such that ω(c ) = β, ω(D(c , 3Aν)) ⊃ D(β, ν). β β Proof. — The uniqueness of c comes from the fact that R t → ω(t) ∈ R is in- creasing. For the existence of c we just notice that if I ⊂] − ρ, ρ[ there is nothing to β β prove (notice that ω is increasing) and otherwise for some ε ∈{−1, 1}|ω(ερ) − β|≤ −1 (7/6)ν.But then,the fact that A ≤ ∂ω(t) ≤ A shows the existence of a unique c ∈[−ρ − (7/6)Aν, ρ + (7/6)Aν] such that ω(c ) = β . β β −1 −1 If we set a = ∂ω(c ) and b = ∂ω(c ) one has a ≥ A , |b|≤ (7A) (cf. (B.459) β β 2 −1 and the fact that ν ∈]0,(6A B) [) and the linear map w → Dω(c )w = aw + bw is invertible, the norm of its inverse being ≤ (7/6)A. Next, we observe that because |ω(c + β 138 RAPHAËL KRIKORIAN w) − ω(c ) − Dω(c )w|≤ B|w| /2and ω(c + w ) − ω(c + w) = Dω(c + w + β β β β β t(w − w))(w − w)dt,the map g −1 −1 g : w → Dω(c ) u − Dω(c ) ω(c + w) − ω(c ) − Dω(c )w β β β b β is (7/6)AB-Lipschitz on {|w|≤ } (some > 0) and sends {|w|≤ } into itself pro- −1 vided (7/6)AB ≤ 1and |u|≤ (6/14)A . In particular if one chooses = 3Aν the 2 −1 map g is (1/2)-contracting (remember ν ∈]0,(6A B) [) and the Contraction Mapping Theoremshows that forany |u|≤ ν ≤ (9/7)ν there exists a unique |w|≤ 3Aν such that ω(c + w) = β + u. We can now prove Proposition 2.7. We ﬁrst observe that the computations done in −1 the proof of Lemma B.2 show by the same token that the map w → Dω(0) (ω (w) − ω(0) − Dω(0)w) is ((7/6)ABρ)-Lipschitz on D(0,ρ) hence contracting from (2.60). This implies that ω is injective when restricted to D(0,ρ) ⊃ U. Assume that (2.61)is satisﬁed for no z ∈ U. Then Lemma B.1 tells us that there exists t + is ∈ Usuch that (B.458) holds and in particular that the interval I of Lemma B.2 is not empty. Applying this latter lemma and using the injectivity of ω when restricted to U we see that ω(U D(c , 3Aν)) ⊂ C D(β, ν) which is the searched for conclusion. Appendix C: Illustration of the screening effect We describe in this section an example where the screening effect mentioned in Sec- tion 3.2 is effective. Consider U as in (3.73) (C.464)U = D(0,ρ) D(c ,ρ ) ⊃ D(0,σ ) j j j=1 (j = 1,..., N, D(c ,ρ ) ⊂ D(0,ρ), ρ σ ), with j j j β γ −N −N ρ = 1,σ = e < 1/10,ρ = ρ := e (j ∈{1,..., N}), j 1 c = (1/4) + (2j − 1)/(4N) ∈[1/4, 3/4]. We assume N 1. The function ϕ(·) := ω (·,∂ D(0,σ )) is harmonic on U D(0,σ ), equal to 1 on ∂ D(0,σ ) and equal UD(0,σ ) to0on ∂ U = ∂ D(0, 1) D(c ,ρ ). Note that the minimum value of the Dirichlet j 1 j=1 integrals |∇ψ(x + iy)| dxdy UD(0,σ ) ψ ∈ H (U D(0, σ )), ψ = 1,ψ = 0 | ∂ D(0,σ ) | ∂ U ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 139 (H denotes the usual Sobolev space) is achieved at ϕ, hence there exists a constant C independent of N such that (C.465) |∇ϕ(x + iy)| dxdy ≤ C; UD(0,σ ) (the constant C is for example the Dirichlet integral of any ﬁxed C function ψ equal to 1on ∂ D(0,σ ) andto0on U D(0, 1/5)). We now use a result by Rauch and Taylor [40] that we adapt to the case of the complex plane: let C be the holed rectangle U ∩ ([1, 4, 3/4]+ −1[−H, H]) with H = C δ ln(δ/ρ ), where δ = 3/(8N), 1 1 C being some large constant. The set C can be covered by the N 1-holed rectangles 1 H U ∩ ([c − δ, c + δ]+ −1[−H, H]), j ∈{1,..., N}, each point of C belonging to no j j H more than two of these holed rectangles. An adaptation of Inequality (4.1) of [40]tothe case of domains in R ( C) asserts that in this situation there exists a constant C > 0 (independent of ϕ, H, N) such that −1 |∇ϕ(x + iy)| dxdy C 2 |ϕ(x + iy)| dxdy H(H + δ ln(δ/ρ ) which in view of (C.465) and the choices made for H and δ implies 2 −(1−γ−) |ϕ(x + iy)| dxdy ≤ CC (C + 1)δ ln(δ/ρ ) N , 2 1 1 because ρ = exp(−N ). In particular, for any μ> 0 there exists a positive constant C 1 μ and a set C of relative measure 1 − μ in C such that for any z ∈ C H,μ H H,μ −(1−γ−μ) |ω (z,∂ D(0,σ ))|=|ϕ(z)|≤ C N , UD(0,σ ) an inequality which is in sharp contrast with (3.71), especially if 1 − γ − μ>β . Getting a useful estimate like (3.72) by this technique is therefore doomed to fail if α< 1− γ − μ. Appendix D: First integrals of integrable ﬂows This section is dedicated to the proof of the following Lemma, on ﬁrst integrals of the integrable ﬂow φ , that was used in the proof of Lemma 5.1. J∇ r Lemma D.1. — Let U be a σ -symmetric open connected set of D and F ∈ O (W ) such σ h,U that (D.466) ∀ t ∈ R, F ◦ φ = F. J∇ r 140 RAPHAËL KRIKORIAN Then, there exists F ∈ O (U) such that on W one has σ h,U F = F ◦ r. Proof. — The Lemma is clear when we are in the (AA) case since the identity ∀ t ∈ R F(θ + t, r) = F(θ , r) clearly implies that F does not depend on θ.Soweconsiderthe (CC)-case. We shallprove that forevery (z,w) ∈ W there exists an open neighborhood V h,U z,w of (z,w) and a holomorphic function f such that F = f ◦ r on V . z,w z,w z,w We consider three cases: 1) If (z,w) = (0, 0) ∈ W . One can write for μ small enough and (z,w) ∈ D(0,μ) , h,U k l F(z,w) = F z w . The identity (D.466) implies M (F) = 0for n = 0hence from k,l n k,l∈N (5.103) one has F(z,w) = M (F)(z,w) = F (zw) and we can choose f (r) = 0 k,k 0,0 k∈N (ir) . k∈N it 2) If zw = 0, with for example w = 0. Then, from (5.106), t → F(0, e w) is holomor- h 1/2 phic with respect to t ∈ R + i]− ln(e ρ /|w|),∞[ and constant on the real axis; it h 1/2 is hence constant on R + i]− ln(e ρ /|w|),∞[. In particular taking t = is, s ∈ R , −s gives F(0, e w) = F(0,w) and by making s →∞ we get F(0,w) = F(0, 0) (notice that (0, 0) ∈ W in that case). The same argument shows that for ( z, w) ∈ W the function h,U h,U −it it h 1/2 h 1/2 t → F(e z, e w) is constant on t ∈ R + i]− ln(e ρ /|w |), ln(e ρ /| z|)[.Now if ( z, w) h 1/2 is close enough to (0,w),inparticularifthere exists 0 < s < ln(e ρ /| z|) such that s −it it s −s 2 |w |/μ < e <μ/| z|, one has with t = is, (e z, e w) = (e z, e w) ∈ D(0,μ) .By(D.466) −it it and point 1), one gets F( z, w) = F(e z, e w) = f (−i zw) . 0,0 3) Otherwise, we can assume that zw = 0. As before, we can argue that the function −it it t → g (t) := F(e z, e w) is constant on the set z,w h 1/2 h 1/2 R + i]− ln(e ρ /|w|), ln(e ρ /|z|)[. −it Any point ( z, w) ∈ W which is close enough to (z,w) is of the form z = e z, w = h,U it e λw, t close to 0 and λ closeto1.Wethushave −it it −1 F( z, w) = F(e z,λe w) = F(z,λw) = F(z, zw z ) = f ( zw) −1 wherewehavedeﬁned f (r) = F(z, irz ). We have thus proven that for each (z,w) ∈ W there exist a neighborhood V h,U z,w and a holomorphic function f such that F = f ◦ r on V .Now if f ◦ r = f ◦ r z,w z,w z,w z,w z ,w on a nonempty open set, the function f and f coincide on a nonempty open set z,w z ,w and thus there exists a holomorphic extension of f of these two functions such z,w,z ,w that f ◦ r = f ◦ r = f ◦ r on V ∩ V . We can now conclude by using the z,w,z ,w z,w z ,w z,w z ,w connectedness of U. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 141 Appendix E: (Formal) Birkhoff Normal Forms Our aim in this Section is to recall the proof of the existence and uniqueness of the formal BNF, Propositions 6.1, 6.2. This is of course a standard topic but we tried to develop here a framework that is convenient for the proof of Lemma 6.3. We mainly concentrate on the (AA)-case since the formalism in the (CC)-case is very similar to the one developed by Pérez-Marco in [36]. E.1 Formal preliminaries. E.1.1 For mal series. — Let A be a commutative ring and A[[X ,..., X ]] (d ∈ N ) 1 d n n 1 d the ring of formal power series a X , a ∈ A,X = X ··· X (for short X = d n n n∈N d (X ,..., X )). We denote by v(A) = min{|n|= n + ··· + n , a = 0} the valuation 1 d 1 d n n d of an element A = a X and if B = (B ,..., B ) ∈ (A[[X]]) we deﬁne v(B) = n 1 d n∈N min v(B ). l l For any k ∈ N we deﬁne [A] = a X the homogenous part of A of degree k k k |n|=k k ∞ and we set [A] = [A] (resp. [A] = [A] ). ≤k l ≥k l l=0 l=k n n As usual the product of A = a X and B = b X is AB = d n d n n∈N n∈N n n ( a b )X and the derivative ∂ A = n a X with n = n if j = l and d d k l X l n j l j n∈N k+l=n n∈N n = n − 1is n ≥ 1(if n = 0 the derivative of the corresponding monomial is zero). Note l l l that if A ∈ A[[X]],1 ≤ i ≤ j one has (E.467) [A ... A ] = [A ] ···[A ] . 1 j k 1 k j k 1 j k +···+k =k 1 j n d When A = a X ∈ A[[X]] and B ∈ (A[[X]]) , v(B) ≥ 1 one can deﬁne d n n∈N A ◦ B = a B . n∈N If moreover A is endowed with derivations δ : A → A,1 ≤ i ≤ d ,(it means δ (a + i i b) = δ a + δ b, δ (ab) = (δ a)b+ a(δ b))wedeﬁne (cf. Taylor formula) for each a ∈ A and i i i i i B ∈ (A[[X]]) , v(B) ≥ 1 k k (E.468) a ◦ B = (1/k!)(δ a)B ∈ A[[X]]. k∈N k d Similarly, if A = a X , B, C ∈ (A[[X]]) , v(B) ≥ 1, v(C) ≥ 1, we can also deﬁne d k k∈N (E.469)A ◦ (B, C) = (a ◦ B)C . δ n δ n∈N k k k k 42 k 1 d k 1 d We use a multi-index notation, k = (k ,..., k ),B = B ··· B , k!= k !··· k !, δ = δ ··· δ . 1 d 1 d 1 1 d d 142 RAPHAËL KRIKORIAN Lemma E.1. — For k ∈ N , v(A) ≥ 1, v(B) ≥ 1, v(C) ≥ 2, [A ◦ (B, X + C) − A] δ k is a polynomial in the coefﬁcients of [δ A] ,[B] ,[C] for k + k + k ≤ k − 1, |l|≤ k (this k k k 1 2 3 1 2 3 polynomial being with rational coefﬁcients). n n n n n Proof. — Since (a ◦ B)(X + C) − a X = (a ◦ B)((X + C) − X ) + X ((a ◦ n δ n n δ n δ B) − a ) A ◦ (B, X + C) − A = (I) + (II) l l n δ a δ a n n l m n−m l n (I) := B C X ,(II) = B X m l! l! |l|≥0 |l|≥1 |n|≥1 |n|≥1 m≤n, |m|≥1 and one can conclude using (E.467). Assume now that (A,δ) is endowed with a translation by which we mean an action τ of an abelian group (we suppose it is (R ,+))on A that commutes with the derivations δ . E.1.2 Formal diffeomorphisms. — A formal diffeomorphism of A[[X]] is a triple (α, A, B) (we denote it by f )with A, B ∈ (A[[X]]) with v(B) ≥ 2and where v(A) ≥ 1and α,A,B α ∈ R . We can deﬁne the composition of two such objects: f = f ◦ f ⇐⇒ ε,E,D γ,C,D α,A,B ε = α + γ, E = A + (τ C) ◦ (A, X + B), −α δ F = B + (τ D) ◦ (A, X + B) −α δ with v(E) ≥ 1and v(F) ≥ 2. One can check that the usual algebraic rules for composi- tions are satisﬁed and that each such diffeomorphism has an inverse for composition. Remark E.1. — One of the example we have in mind is the following. Take d = ∗ ω d d d ∈ N , A = C (T ) the ring of real analytic functions on T (taking real values on the n ω d real axis) and the ring of formal power series is A[[r]] = { a (θ )r , a ∈ C (T )}, n n n∈N r = (r ,..., r ). The derivations in this case are δ a = ∂ a if a : (θ ,...,θ ) → R is in 1 d i θ 1 d ω d d C (T ), the translation is τ a = a(·− α) (α ∈ R ) and the formal map f can be α α,A,B written under the more suggestive form f (θ , r) = (θ + α + A(θ , r), r + B(θ , r)) (α,A,B) d d as a formal diffeomorphism of T × R . ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 143 E.1.3 Degree. — In case we can assign a degree deg(a) to each element a of the ring A (it satisﬁes by deﬁnition deg(0) =−∞,for all a, b ∈ A,deg(a + b) = max(deg(a), deg(b)) and deg(ab) = deg(a) + deg(b)) we can associate to each weight p : N → N the set n d (E.470) C(p) = a X ∈ A[[X]], ∀ n ∈ N , deg(a ) ≤ p(n) . n n n∈N By extension if B = (B ,..., B ) ∈ (A[[X]]) we say that B is in C(p) if each B ∈ C(p), 1 d l 1 ≤ l ≤ d .If p, q : N → N we deﬁne p ∗ q(n) = max (p(k) + q(l)). (k,l)∈N ,k+l=n In particular if (E.471) p(n) := |n|= n +···+ n , n = (n ,..., n ) ∈ N 1 d 1 d ∗m one has p ∗ p = p and (p − 1) = p − m. We say that the degree deg is compatible with the derivations δ and the translation τ if for any α ∈ R ,1 ≤ i ≤ d ,deg(τ δ a) ≤ deg(a). α i Remark E.2. — The relevant example for our purpose (proof of Lemma 6.3)will ω d be the following. Take d = d , A = C (T )[t] the set of polynomials in t with coefﬁcients ω d (t) n ω d in C (T ),F (θ ) = a (θ ) + ··· + a (θ )t , a ∈ C (T ),0 ≤ j ≤ n, n ∈ N and a = 0. 0 n j n (t) n The derivations δ ,1 ≤ i ≤ d are deﬁned by δ F (θ ) = (∂ a )(θ ) + ··· + (∂ a )(θ )t , i i θ 0 θ n i i (t) n t the translations τ F (θ ) = a (θ − α) + ··· + a (θ − α)t and the degree deg F = n is α 0 n compatible with both of them. The following facts are easily checked. Assume that A is a ring with derivations δ , 1 ≤ i ≤ d and a compatible degree deg and let (p ) be weights. l l∈N (1) If A, B ∈ A[[X]],A ∈ C(p ),B ∈ C(p ) one has AB ∈ C(p ∗ p ). 1 2 1 2 (2) If A ∈ A[[X]],A ∈ C(p ),lim v(A )=∞ one has A ∈ C(max p ). l l l l→∞ l l l l l∈N Let p be a weight such that p ∗ p ≤ p. Using (E.468), points (1)and (2)wehave (3) If a ∈ A,B ∈ A[[X]],B ∈ C(p) then one has a ◦ B − a ∈ C(deg(a) + p). Lemma E.2. — If A ∈ A[[X]], B, C ∈ (A[[X]]) , v(B) ≥ 1, v(C) ≥ 1 with A ∈ C(p− c ), B ∈ C(p− c ), C ∈ C(p− 1), min(c , c ) ≥ 0,then A◦ (B, C)− A◦ C ∈ C(p− c − c ) A B A B δ A B and A ◦ (B, C) ∈ C(p − c ). δ A Proof. — Recall that A ◦ (B, C) = (a ◦ B)C . From point (3) a ◦ B − a ∈ δ n δ n δ n n∈N n n C(deg(a ) + p − c ) and from point (1) (a ◦ B)C − a C ∈ C((deg(a ) + p − c ) ∗ (p − n B n δ n n B ∗|n| n n 1) ) ⊂ C((deg(a ) + p − c ) ∗ (p−|n|)) hence (a ◦ B)C − a C ∈ C(p − c − c ).By n B n δ n A B 144 RAPHAËL KRIKORIAN using point (2)wehave A ◦ (B, C) − A ◦ C ∈ C(p − c − c ). A similar argument shows δ A B that A ◦ C ∈ C(p − c ) whence the conclusion. Before stating the next lemma we introduce the following deﬁnition: we say that a formal diffeomorphism f is in D(p − 1) if A ∈ C(p) ∩ O(r),B ∈ C(p − 1) ∩ O (r). α,A,B Lemma E.3. — One has (1) Let H ∈ C(p − c),c = 0, 1,and f ∈ D(p − 1).Then H ◦ f ∈ C(p − c). α,A,B α,A,B (2) The composition of two formal diffeomorphisms in D(p − 1) is in D(p − 1). (3) The inverse for the composition of a diffeomorphism of D(p − 1) is in D(p − 1). −1 (4) If f = f , then for any k ≥ 1, [A] , [B] , are polynomials in the coefﬁcients of k k α,A,B α,A,B l l 1 2 [τ δ A] ,[τ δ B] ,k , k ≤ k, l , l ≤ k, |m |,|m |≤ k. m α k m α k 1 2 1 2 1 2 1 1 2 2 Proof. — Items 1 and 2 are consequences of Lemma E.2. For point 3 we just have to prove the result when α = 0. Let us denote by U the operator H → H ◦ f .Notethat v((U − id)H) ≥ v(H) + 1 hence the series H := 0,A,B (U − id) Hconverges in A[[r]] and from 1 and 2 one sees that if H ∈ C(p − c), k=0 −1 c = 0,1, thesameistrue for H. To conclude we observe that −(f − id) = (U − 0,A,B −1 id)(f − id) + (f − id) hence 0,A,B 0,A,B −1 k (E.472) −(f − id) = (U − id) (f − id). 0,A,B 0,A,B k=0 Finally point 4 is a consequence of (E.472) and Lemma E.1. E.2 Formal Birkhoff Normal Forms. — From now on we work in the setting of Re- mark E.2. E.2.1 Formal exact symplectic diffeomorphism. — If F ∈ A[[r]],F=α, r+ O (r) with α ∈ R we deﬁne the formal diffeomorphism f (θ , r) = (θ + A(θ , r), r + B(θ , r)) as sug- gested by the implicit relation ϕ = θ + ∂ F(θ , R), r = R + ∂ F(θ , R), f (θ , r) = (ϕ, R) R θ F or more formally A(θ , r) = ∂ F(θ , r + B(θ , r)), 0 = B(θ , r) + ∂ F(θ , r + B(θ , r)) r θ A = ∂ F ◦ (0, B), 0 = B + ∂ F ◦ (0, B)(cf. E.469).) r δ θ δ Note that to prove the existence and uniqueness of the formal BNF of Section E.2.2 it would be enough to work ω d with A = C (T ). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 145 In this situation we use the more intuitive notations R(θ , r) = r + B(θ , r), ϕ(θ, r) − θ = A(θ , r). We shall call such formal diffeomorphisms f formal exact symplectic diffeomorphisms. The set of all such diffeomorphisms is a group under composition. Let us deﬁne E (p − 1) := {f , F=α, r+ F (θ )r ∈ A[[r]], F ∈ C(p − 1)}. F n |n|≥2 The following result is then a consequence of Lemma E.3. Lemma E.4. — The set E (p− 1) is a subset of D(p− 1) stable by composition and inversion. Remark E.3. — In the (CC)-case the relevant choice for A is C[[t]] and the set of formal series is A[[z,w]]. One can extend in this context the notion of σ - m m symmetry. If F = F z w ∈ A[[z,w]] we say it is σ -symmetric (recall d d n,m 2 (n,m)∈N ×N |n|+|m| d d σ (z,w) = (iw, iz))if F = (i) F for all (n, m) ∈ N × N (F is the complex 2 n,m m,n n,m conjugate of F and i = −1). Similarly one can deﬁne the notion of σ -symmetric n,m 2 formal diffeomorphism. If F is σ -symmetric then f is σ -symmetric. 2 F 2 E.2.2 Existence and uniqueness of the formal BNF. — We prove at the end of this sub- section that given f we can ﬁnd B(r) = 2πω , r+ O (r) ∈ R[[r]] and a formal 2πω ,r+F 0 2 2 exact symplectic diffeomorphism f = id + O (r),Z = O (r) which is normalized in the sense that (E.473) Z(ϕ, Q)dϕ = 0, such that (E.474) f ◦ f (θ , r) = f ◦ f (θ , r). Z 2πω ,r+F B Z Moreover, Z and B are uniquely determined by F. We use the notations f : (θ , r) → (ϕ, R), f : (ϕ, R) → (ψ, Q) so that 2πω ,r+F Z ψ = ϕ + ∂ Z(ϕ, Q), R = Q + ∂ Z(ϕ, Q) Q ϕ ϕ = θ + 2πω + ∂ F(θ , R), r = R + ∂ F(θ , R). 0 R θ Using the relation R = Q + ∂ Z(θ + 2πω + ∂ F(θ , R), Q) and the fact that ϕ 0 R g : (θ , Q, R) → (θ , Q, R − Q − ∂ Z(θ + 2πω + ∂ F(θ , R), Q)) is a formal diffeomor- ϕ 0 R 44 2 −1 phism we can deﬁne R(θ , Q) = Q + O (Q) by (θ , Q, R(θ , Q)) = g (θ , Q, 0). Deﬁned on A[[Q, R]]. 146 RAPHAËL KRIKORIAN Lemma E.5. (1) For any k ≥ 1,the coefﬁcientsof [R(θ , Q)] are polynomials in the coefﬁcients of [τ δ F] , k 2π m ω k 1 0 1 [τ δ Z] ,k , k , l , l ,|m |,|m |≤ k. 2π m ω k 1 2 1 2 1 2 2 0 2 (2) If F, Z ∈ C(p − 1), the formal diffeomorphism (θ , Q) → (θ , R(θ , Q)) is in D(p − 1). Proof. — These are consequences of Lemma E.3. Let f (θ , r) = (θ , r ); from the formal conjugation relation (E.474)weget f ◦ Z B f (θ , r) = (θ +∇ B(r ), r ) = (ψ, Q) hence Q = r and θ = θ + ∂ Z(θ , Q).Wethus Z Q have θ + ∂ Z(θ , Q)+∇ B(Q) = ϕ + ∂ Z(ϕ, Q) Q Q and using the relations between ϕ, θ yields −∂ F(θ , R) = ∂ Z θ + 2πω + ∂ F(θ , R), Q − ∂ Z(θ , Q) R Q 0 R Q − (∇ B(Q) − 2πω ) that we can write (E.475) −∂ F (F, Z) = ∂ Z(θ + 2πω , Q) − ∂ Z(θ , Q) − (∂ B(Q) − 2πω ) Q Q 0 Q Q 0 where F (F, Z) = O (r) is uniquely deﬁned (note that the RHS of (E.475)is O(r))by (E.476) ∂ F (F, Z) = ∂ F(θ , R(θ , Q)) Q Q + ∂ Z θ + 2πω + ∂ F(θ , R(θ , Q)), Q − ∂ Z(θ + 2πω , Q) . Q 0 Q Q 0 We thus have (E.477) −F (F, Z) = Z(θ + 2πω , Q) − Z(θ , Q) − (B(Q) − 2πω , Q). 0 0 Lemma E.6. (1) For any k ≥ 1,the coefﬁcientsof [F (F, Z) − F] are polynomials in the coefﬁcients of l l 1 2 [τ δ F] ,[τ δ Z] ,k , k ≤ k − 1,l , l ≤ k, |m |,|m |≤ k. 2π m ω k 2π m ω k 1 2 1 2 1 2 1 0 1 2 0 2 (2) If F, Z ∈ C(p − 1), one has F (F, Z) ∈ C(p − 1). Proof. — This is a consequence of (E.476), Lemma E.5 and Lemma E.3. From (E.477) one thus has k = 2, −[F] (θ , Q)=[Z] (θ + 2πω , Q)−[Z] (θ , Q)−[B] (Q), 2 2 0 2 2 (E.478) ∀ k ≥ 3 −[F (F, Z)] (θ , Q)=[Z] (θ + 2πω , Q)−[Z] (θ , Q)−[B] (Q). k k 0 k k ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 147 Before completing the proof of the existence and uniqueness of the Birkhoff Nor- mal Form (E.474) we state the following result (the ﬁrst part of which at least is classical; see for example [13]): Lemma E.7. — If ω is Diophantine, for any G ∈ A[[r]] there exists a unique pair (Z, B) with Z ∈ A[[r]] normalized in the sense of (E.473)and B = B(r) ∈ R[t][[r]] such that (E.479)G(θ , Q) = Z(θ + 2πω , Q) − Z(θ , Q) + B(Q). Furthermore: (1) B(Q) = G(θ , Q)dθ and if G =[G] one has Z =[Z] , B =[B] and d k k k the coefﬁcients of [Z] are R-linear functions of the coefﬁcients of [G] ;(2) if G ∈ C(p − 1) then k k Z, B ∈ C(p − 1). −d −il,θ Proof. — If we denote by G(l, Q) = (2π) G(θ , Q)e dθ the l -th Fourier coefﬁcient of θ → G(θ , Q) (l ∈ Z )then(E.479) follows if B(Q) = G(θ , Q)dθ and if Z(θ , Q) is deﬁned by 2π il,ω −1 il,θ Z(θ , Q) = (e − 1) G(l, Q)e . l∈Z {0} The conclusions of the Lemma are clear from the preceding expression. Proof of the existence and uniqueness of the BNF (E.474). – Uniqueness: Equation (E.478), Lemma E.6 and Lemma E.7 show inductively that [Z] ,[B] are uniquely deﬁned by [F] ,2 ≤ j ≤ k − 1. Hence, Z and B are unique. k k j –Existence:Deﬁne [Z] ,[B] by (E.478) and then inductively for k ≥ 3, [Z] ,[B] ,by 2 2 k k (E.480) −[F (F,[Z] )] (θ , Q)=[Z] (θ + 2πω , Q)−[Z] (θ , Q)−[B] (Q) ≤k−1 k k 0 k k k−1 ∞ ∞ where Z = [Z] . Setting F = [Z] ,B = [B] one can check that (E.477) ≤k−1 l l l l=2 l=2 l=2 holds modulo O (r) for any k and hence in A[[r]]. (t) E.3 Proof of Lemma 6.3.— We deﬁne F (θ , r) = tF(θ , r) + (1 − t)G(θ , r) which ω d (t) is in A[[r]] ∩ C(p − 1), A = C (T )[t]. Note that for any k ≥ 2, [F ] ∈ C(p − 1).In particular, as a consequence of Lemmata E.6,item 2 and E.7, point (2), the sequences (t) (t) (t) [Z ] ,[B ] , inductively constructed in (E.480), are in C(p − 1).Hence B (r) := k k b (t)r is in C(p− 1) which by deﬁnition (cf. (E.470), (E.471)) means that the degree d n n∈N in t of each b (t) is ≤|n|− 1. Appendix F: Approximate Birkhoff Normal Forms We give in this section the proofs of Propositions 6.4 and 6.5. 148 RAPHAËL KRIKORIAN F.1 A useful lemma. — Let be given for each α ∈]0, 1/2[,afunction P : R × α + R → R ,P : (k, t) → P (k, t) nondecreasing in each variable and assume that + + α α (ε ) ,I ⊂ N is a sequence of nonnegative real numbers depending on α ∈]0, 1/2[ α,k k∈I α and deﬁned inductively as long as a condition of the form (F.481)P (k,ε )< 1 α α,k ∗ ∗ is satisﬁed (we assume that ε satisﬁes (F.481)). Let us call J = 0, k , k ≥ 1the maxi- α,0 α α α mal set of integers k ∈ N for which ε is deﬁned: this means that if k ∈ J and ε satisﬁes α,k α α,k ∗ ∗ (F.481)then k + 1 ∈ J (in particular P (k ,ε ) ≥ 1). Let θ> 0, a > 0and k ∈ N be α α α,k θ,a α α such that ∗ θ −a (F.482) ∀k ∈ 0, min(k , k ) − 1,ε ≤ C α × 1 + α ε ε . θ,a α,k+1 θ,a α,j α,k j=0 We have the following type of Gronwall Lemma: Lemma F.1. — Assume that θ a (F.483) (2C α) < 1/2,ε ≤ α /2, P (k + 1,ε )< 1. θ,a α,0 α θ,a α,0 Then, (F.484) k ≥ k θ,a and θ k (F.485) ∀ k ∈[0, k ]∩ N,ε ≤ (2C α) ε . θ,a α,k θ,a α,0 ∗ ∗ Proof. — 1) Let k = min(k , k ). We ﬁrst prove that the set θ,a α,θ ,a α ∗ θ k K ={k ∈ 0, k ,ε >(2C α) ε } α,θ ,a α,k θ,a α,0 α,θ ,a is empty. If this were not the case we could deﬁne k = inf K and write α,θ ,a α,θ ,a θ k (F.486) ∀ k ∈ 0, k − 1,ε ≤ (2C α) ε α,θ ,a α,k θ,a α,0 hence α,0 ε ≤ ≤ 2ε α,j α,0 1 − (2C α) α,θ ,a j=0 and thus from (F.482)and (F.483), for all 0 ≤ k ≤ k − 1, α,θ ,a θ −a θ ε ≤ C × α (1 + 2α ε )ε ≤ (2C α )ε . α,k+1 θ,a α,0 a,k θ,a α,k ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 149 This implies that for all 0 ≤ k ≤ k one has α,θ ,a θ k ε ≤ (2C α) ε α,k θ,a α,0 θ k α,θ ,a and in particular ε ≤ (2C α) ε . This contradicts the deﬁnition of k as α,k θ,a α,0 α,θ ,a α,θ ,a inf K . α,θ ,a 2) Since K is empty, one has α,θ ,a ∗ kθ (F.487) ∀ k ∈ 0, k ,ε ≤ (2C α) ε ≤ ε . α,k θ,a α,0 α,0 α,θ ,a But since k + 1 ≤ k + 1and (2C α) ≤ 1 θ,a θ,a α,θ ,a P (k + 1,ε ) ≤ P (k + 1,ε ) α α,k α θ,a a,0 α,θ ,a α,θ ,a < 1 ∗ ∗ ∗ ∗ This implies that k + 1 ∈ J hence k < k and from the deﬁnition of k ,weget α,θ ,a α,θ ,a α α,θ ,a ∗ ∗ k ≥ k which in its turn implies k = k .Wehavethusproventhatfor all k ≤ k θ,a θ,a θ,a α α,θ ,a one has kθ ε ≤ (2C α) ε . α,k θ,a α,0 F.2 Proof of Proposition 6.4 (BNF, (CC)-Case). — For n large enough we deﬁne (F.488) ρ = , W = e W 0 0 h,D(0,ρ ) and for k ≥ 0 we introduce the sequences −1 (F.489) δ = C , δ ≤ h/10 k l (k + 1)(ln(k + 2)) l=0 k−1 k−1 (F.490) ρ = exp(− δ )ρ , W = exp(− δ )W . k l 0 k l 0 l=0 l=0 Recall that a = max(2a + 1, 30) and that for some m ≥ a (cf. (6.141)) 1 1 2m m (F.491)F (z,w) = O (z,w), ε := F ρ . 0 0 0 W The choice of these sequences, in particular the choice of a summable sequence δ , is not necessary (we only perform a ﬁnite number of conjugation steps) and is indeed not the best one insofar as it leads to a(n) (arbitrary small) loss β in the exponent. The “optimal” one is (after loosing a ﬁxed fraction of h in the ﬁrst step) to choose uniformly at each ∗ ∗ step δ ∼ h/k so that δ ∼ h where k is the ﬁnite number of conjugation steps we perform. k ∗ k 0≤k≤k 150 RAPHAËL KRIKORIAN We shall construct inductively for k ≥ 0, sequences Z ∈ O (W ),F ∈ O (W ) ∩ k σ k k σ k k+2m 2 O (z,w), ∈ O (D(0,ρ )), (r) = 2πω r + O (r),suchthat k σ k k 0 = , F = F 0 0 and for k ≥ 1, −1 (F.492) g ◦ ◦ f ◦ g = ◦ f . F k F k 0 0 k k −1 To do this we proceed the following way: assuming (F.492) holds and δ ≤ q ,weap- ply Proposition 5.3 with τ = 0, K/2 = N = q cf. (6.143), and deﬁne Y ∈ O (W ) ∩ n k σ k k+2m O (z,w) (see Remark 5.3) satisfying (F.493) −[Q ]· Y = T F − M (F ), Y −δ /2 q F 0 k q k 0 k k k k W n e W k k n where we denote (F.494)Q (r) = 2πω r. 0 0 Using Lemma 5.4 we get (cf. formula (5.124)) −1 f ◦ ◦ f ◦ f = ◦ f (a) Y F +M(F ) ˙ k k k Y k k k F −M (F )+[ +M(F )]·Y +O (Y ,F ) k 0 k k k k k k = ◦ f (a) . +M(F ) ˙ k k R F +[ +M(F )]·Y −[Q ]·Y +O (Y ,F ) q k k k k 0 k k k n 2 −1 Hence f ◦ ◦ f ◦ f = ◦ f with Y F F k k k Y k+1 k+1 (F.495) − = M(F ) k+1 k k and, using the fact that [ + M(F )]· Y −[Q ]· Y = O(|∇ Y |×|∂( − Q ) ◦ r|) + k k k 0 k k k 0 (2) O (Y , F ), k k (a) (F.496)F = R F + O (Y , F ) + O |∇ Y |×|∂( − Q ) ◦ r| . k+1 q k k k k k 0 n 2 j+2m Notice that from (F.495) and the fact that for 0 ≤ j ≤ k − 1, M (F ) = O (z,w) (cf. the 0 j remark at the end of Section 5.1.1), hence M(F)(r) = O (r); one thus has (F.497) ∀ 0 ≤ j ≤ k, (r) − Q (r) = O(r ). k 0 (a) k+2m 2k+4m−2a Since F , Y ∈ O (z,w) we have (see Remarks 2.1, 5.3) O (Y , F ) = O (z,w); k k k k k+2m+1 q also, O(|∇ Y|×|∂( − Q ) ◦ r|) = O (z,w) and from (5.115)R F = O (z,w) k 0 q k (if q ≥ m). As a consequence, since 2k + 4m − 2a ≥ k + 1 + 2m (m ≥ 2a + 1) we see that n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 151 k+1+2m 2 F = O (z,w). Furthermore since (r) − Q (r) = O(r ) k+1 0 0 ∂( − Q ) ◦ r ≤∂( − ) ◦ r +∂( − Q ) ◦ r k 0 W k 0 W 0 0 W k k k ∂( − ) ◦ r + sup|r(W )| k 0 W k ∂( − ) ◦ r + ρ k 0 W k and using (2.55) −1/2 −1 ∇ Y −δ δ ρ Y −δ /2 k k k k e W k e W k k k −1/2 2 −1 q ρ δ F k W k k n k hence 1/2 2 −2 −2 2 −1 (F.498) |∇ Y |×|∂( − Q | q ρ δ − F + q δ ρ F . k k 0 k 0 D(0,ρ ) k W k W k k k k n k k n k From (F.496), (F.493), (F.498), and (5.112) we get that, provided −a −a 2 ρ δ q F < 1, k W k k n one has the inequalities −1 −q δ /2 −a −a 2 2 n k −δ (F.499) F δ e F + ρ δ q F k+1 k k W k e W k k k k k n W 1/2 2 −2 −2 2 −1 + q ρ δ − F + q δ ρ F k 0 D(0,ρ ) k W k W n k k k k n k k k and (F.500) − −δ F . k+1 0 D(0,e ρ ) j W j=1 Let us deﬁne s = − ,ε =F ; k k 0 D(0,ρ ) k k W k k then, one has 1/2 ⎨ −1 −q δ /2 −a 2 2 −2 2 −1 ε δ e + (ρ δ ) q ε + q (ρ δ ) s + q δ ρ ε k+1 k k k k k k k k k k n n n (F.501) s ε k+1 j j=0 as long as −a −a 2 ρ δ q ε < 1. k k n 152 RAPHAËL KRIKORIAN Let k be the largest integer for which the preceding sequences are deﬁned and satisfy the stronger condition −1/5 ∗ −a −a 2 (F.502) ∀ k < k , P (k,ε ) := ρ ρ δ q ε < 1. q k k n 0 k k n −(1+σ) From (F.489)for any σ> 0 one has δ (k + 1) .For θ ∈]0, 1/6[ deﬁne k σ 1 1 μ = − θ, μ = μ . 0 0 6 1 + σ Since ρ ρ one has k 0 −1/6+μ 0 −θ q δ ρ = ρ n k ⎨ 0 0 ∗ −μ −1−μ −7/6+θ −1 0 ∀ k < min(k ,ρ ), (ρ δ ) ρ = ρ k k 0 0 −1 1/2 −1/3+1/2−μ 2 0 θ q δ ρ ρ = ρ , k k 0 n 0 −μ 1/5 −1 −q δ /2 n k thus (note that for k <ρ , ρ 1), δ e ≤ ρ and consequently, if ρ 1, 0 θ 0 θ 0 k 0 1/5 1/5 −1/3−7/3+2θ ∗ −μ θ ∀ k < min(k ,ρ ), ε (ρ + ρ + ρ ε + ρ )ε k+1 j k 0 0 0 0 0 j=0 θ −3 ≤ C ρ (1 + ρ ε )ε . θ j k 0 0 j=0 1/5+(7/6)a+1/3 2a+1 Since from (F.491) ε ≤ ρ <ρ we see that condition (F.483) of Lemma F.1 0 0 −μ is satisﬁed (with α = ρ , k = ρ )hence 0 θ,α −μ θ k (F.503) ∀ k ∈[0,ρ ]∩ N,ε ≤ (2C ρ ) ε . k θ 0 0 −μ 1−β Now for any 0 <β 1, one can choose θ and σ so that ρ = q and in particular 0 n 1−β taking k = k =[q ] and using (F.501)one gets for q 1 β n β 1−β −q (F.504) ε ≤ e , s ≤ 2ε k k 0 (we can assume a > 10). We now deﬁne BNF BNF F = F, = , −1 −1 k k q q n n and BNF −1 −1 g = f ◦···◦ f −1 Wh Wh q Y Y k ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 153 Wh 2 −δ /2 where Y is a C Whitney extension of (Y , e W ) given by Lemma 2.2; one has (cf. j j (F.493), (F.491)) BNF 2 θ k −4 −4 8 −(m−26) (F.505) g − id 1 q (2C ρ ) ε ρ h k q . −1 C θ 0 0 θ,h n 0 n k=0 k+2m Inequalities (F.504)and (F.505), and the fact that F ∈ O (r), give the conclusion 1−β BNF q of Proposition 6.4.Notethat(6.145) is a consequence of F ∈ O (z,w) and −1 Remark 6.2. For the last statement of the Proposition, we can choose = + k−1 Wh Wh 3 M(F ) where F is an C Whitney extension of (F , W ) given by Lemma 2.2. j j j j=0 J F.3 Proof of Proposition 6.5 (BNF (AA) or (CC) Case, ω Diophantine). — The proof, that we mainly illustrate in the (AA)-Case, as well as the notations, are essentially the same as the ones of the proof of Proposition 6.4 (see Section F.2, in particular, we use the deﬁnitions (F.489), (F.490)for δ , ρ ,W ) with the following differences: k k k – we replace (F.488)by (τ+1)/γ ρ = ρ , W = W = W bτ 0 0 h,D(0,ρ ) h,D(0,ρ ) where γ = 1 in the (AA)-case and γ = 1/2 in the (CC)-case. – since ω is Diophantine, we can and do solve instead of (F.493) the equation without −1 truncation (using Proposition 5.3 with N=∞,K = κ , cf. (6.149)): −τ −[Q ]· Y = F − M (F ), Y −δ /2 δ F 0 k k 0 k k e k W k W k k where Q (r) = 2πω r,cf. (F.494). 0 0 Notice that both in the (AA) and (CC) cases m m (F.506)F = O (r), F ≤ ρ . 0 0 W 0 0 In place of (F.496) we get (in the (AA)-case) (a) (F.507)F = O (Y , F ) + O |∂ Y |×|∂( − Q ) ◦ r| . k+1 k k θ k k 0 k+m Since F , Y ∈ O (r)(θ ) (see Remarks 2.1, 5.3)wehave O(|∂ Y |×|∂( − Q )◦ r|) = k k θ k k 0 (a) k+m+1 2k+2m−a O (r)(θ ) and O (Y , F ) = O (r)(θ ) (a from Lemma 5.4). As a consequence, k k k+1+m since 2m ≥ a we see that F = O (r)(θ ). k+1 From (F.507) and the fact that (cf. (2.54)) −1 ∂ Y −δ δ Y −δ /2 , ∂( − Q ) ◦ r ≤ ρ θ k k k k 0 0 W k e W k e W k k k 154 RAPHAËL KRIKORIAN hence −(1+τ) |∂ Y |×|∂( − Q ) ◦ r| δ ρ F θ k 0 0 W k k k where γ = 1 in the (AA)-case. A similar computation (cf. (F.498)) shows that one can take γ = 1/2 in the (CC)-case. With the notations s = − , ε =F ,wethen k k 0 D(0,ρ ) k k W k k get −(1+τ) γ −(τ+a) −(2+τ) ε (ρ δ ) ε + (ρ δ ) s + δ ρ ε k+1 k k k k k k k k k (F.508) s ε k+1 j j=0 provided −(τ+a) (F.509) (ρ δ ) ε < 1. k k k Let k be the largest integer for which the preceding sequences are deﬁned and satisﬁes −γ −(τ+a) (F.510) ∀k < k , P (k,ε ) := ρ (ρ δ ) ε < 1; ∗ ρ k k k k 0 0 the condition involved in (F.510) implies (F.509). From (F.489)for any σ> 0 one has −(1+σ) δ (k + 1) .Fix θ ∈]0,γ[ and deﬁne k σ γ − θ 1 μ = . 1 + τ 1 + σ Since ρ ρ one has k 0 −1−μ(1+σ) −1 (ρ δ ) ρ −μ k k ∗ 0 ∀ k < min(k ,ρ ), 0 −(1+τ) γ γ−(1+τ)μ(1+σ) δ ρ ρ = ρ . k k 0 0 If we set γ − θ a = 1 + (2 + τ) + θ ≤ (3/2)(2 + τ) + 1 1 + τ we then get using (F.510)and (F.508) −μ γ ∗ −a+θ θ ∀ k < min(k ,ρ ), ε ≤ C ρ + ρ ε + ρ ε k+1 σ j k 0 0 0 0 j=0 θ −a ≤ C ρ × 1 + ρ ε ε . σ j k 0 0 j=0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 155 We now apply Lemma F.1 with α = ρ : since (a > 10) γ − θ max a,(1 + )(τ + a) + 1 ≤ 2(τ + a) ≤ a ≤ m; 1,τ 1 + τ −μ condition (F.506)shows that (F.510) is satisﬁed with k = min(k ,[ρ ]) as well as condi- −μ tions (F.483)for ρ 1. We thus get if k := [ρ ], 0 θ,σ θ k (F.511) ∀ k ∈[0, k]∩ N,ε ≤ ρ ε . k 0 Since θ and σ can be taken arbitrarily close to 1, for any 0 <β 1 one has for ρ 1 −(1−β) −ρ ε ≤ e , s ≤ 2ε . k k We conclude like in the proof of Proposition 6.4 (Section F.2) by deﬁning BNF BNF BNF −1 −1 F = F, = , g = f ◦···◦ f . k k Wh Wh ρ ρ ρ Y Y 1−β k+a BNF (1/ρ) 1,τ Note that since F ∈ O (r) one has F ∈ O (r) and (6.154) is a consequence of Remark 6.4. Appendix G: Resonant Normal Forms In this section we shall only consider the (AA)-Case. Let c ∈ D(0, 1), ρ> 0, h > 0and h h (G.512) ∈ O (e D(c, ρ)) and F ∈ O (e W ) σ σ h,D(c,ρ) where satisﬁes the twist condition (A, B ≥ 1) −1 −1 2 −1 3 ∀ r ∈ R, A ≤ (2π) ∂ (r) ≤ Aand (2π) D ≤ B. Our aim in this section is to give an approximate Normal Form for ◦ f in a neighbor- hood of a q-resonant circle by which we mean that for some (p, q) ∈ Z × N , p ∧ q = 1 −1 (2π) ∂(c) = . This Normal Form is quite similar in spirit (and in its construction) to the approximate BNF. It is used in the paper in Sections 8 (approximate Hamilton-Jacobi Normal Form) and 15 (creating hyperbolic periodic points). As usual we deﬁne ω(c) := ∂(c). 2π 156 RAPHAËL KRIKORIAN Proposition G.1 (q-resonant Normal Form).— There exists a universal constant a ≥ 10 (not depending on q and that we can assume in N) such that, if one has −8 ρ< (Aq) (G.513) F < ρ , h,D(c,ρ) −1/q then the following holds: There exist ∈ O (D(c, e ρ)) ∩ TC(2A, 2B), res cor −1/q −1/q F , F ∈ O (e W ), g ∈ Symp (e W ) σ h,D(c,ρ) RNF h,D(c,ρ) ex,σ such that −1 res cor g ◦ ◦ f ◦ g = ◦ ◦ f ◦ f F RNF 2π(p/q)r F RNF (G.514) res res F is 2π/q − periodic, M (F ) = 0, where ⎪ −1/q − ( − 2π(p/q)r) F D(c,e ρ) W ⎨ h,D(c,ρ) res (G.515) −1/q F F e W h,D(c,ρ) h,D(c,ρ) −2 a −5 g − id 1 (qρ ) F ≤ ρ RNF C h,D(c,ρ) and cor −1/4 (G.516) F −1/q exp(−ρ )F . e W W h,D(c,ρ) h,D(c,ρ) We give the proof of this Proposition in the next subsections. Remark G.1. — The implicit constants in the symbol of the preceding estimates depend on h;if h > 0, they can be bounded above by a constant C whenever h ≥ h . 0 h 0 G.1 A preliminary Lemma. Lemma G.2. — 1) For any (k, l) ∈ Z × Z one has ⎨ either q|kand p|l (G.517) p 1 ⎩ or |k × − l|≥ . q q 2) Let −1 N = (qρA) . For any r ∈ D(c, ρ) and any (k, l) ∈ N × Z, 1 ≤ k ≤ N, which is not in (q, p)Z one has (G.518) |kω(r) − l|≥ 1/(2q). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 157 Proof. — 1) Indeed |k(p/q) − l|=|kp − lq|/q and if the integer kp − lq is 0 then q|k and p|l . 2) We just notice that |kω(r) − l|≥|kω(c) − l|− k|ω(r) − ω(c)| ≥ (1/q) − N∂ω ρ D(c,ρ) ≥ 1/(2q). Deﬁne (see Section 5.1) q−res q−nr q−res (G.519)T F = M (F), T F = T F − T F. k N N N N k∈Z |k|<N q|k q−res q−nr res nr We shall often use in what follows the shortcuts T and T for T ,T . N N N N From (5.101), (5.111) we see that res 2π/q res (G.520)T F ◦ φ = T F N r N and res −1 (G.521) T F −δ δ F . e h,D(c,ρ) h,D(c,ρ) Corollary G.3. — For any F ∈ O (W ), there exists Y ∈ O (W ) such that σ h,D(c,ρ) σ h,D(c,ρ) M (Y) = 0 and nr (G.522)T F=[]· Y. This Y satisﬁes for any 0 <δ < h −1 (G.523) Y −δ qδ F . e W W h,D(c,ρ) h,D(c,ρ) Proof. — This is a simple adaptation of the proof of Proposition 5.3 (the non- resonance condition is replaced by (G.518)). G.2 Elimination of non-resonant terms. Proposition G.4. — There exists a universal constant a (not depending on q) such that if −1 N = (qρA) and 1/8 −1 ρ <(qA) (G.524) F < ρ , h,D(c,ρ) 158 RAPHAËL KRIKORIAN res nr −1/q −1/q then there exist F , F ∈ O (e W ),g ∈ Symp (e W ) such that σ h,D(0,ρ) h,D(0,ρ) ex,σ −1/q −1 (G.525) [e W ] g ◦ ◦ f ◦ g = ◦ f res nr h,D(0,ρ) F F +F res res (G.526)F = T (F + O(qρF )) N h,D(c,ρ) res F being 2π/q-periodic and −2 2 a −5 (G.527) g − id 1 (qρ ) F ≤ ρ C h,D(c,ρ) nr −1/3 (G.528) F −1/q q exp(−ρ )F . e W h,D(c,ρ) h,D(c,ρ) Proof. — Note that we can assume, using Lemma 2.2 that F ∈ O (W ) and σ h,D(c,ρ) satisﬁes a a −7 3 3 −1/(10q) 3 (G.529) ε := F ≤ ρ , F ρ 0 C e W h,D(c,ρ) where a will be deﬁned in (G.539). −1 −1/(10q) Let N = (qρA) .Wedeﬁne W = e W , ρ = ρ and for k ≥ 1 0 h,D(c,ρ ) 0 k−1 k−1 −1 (5q) (G.530) δ = ,ρ = exp(− δ )ρ, W = exp(− δ )W k k l k l 0 4/3 (k + 1) l=0 l=0 nr res nr res and we construct sequences Y , F , F , F ∈ O (W ) such that F = F + F k k σ k k k k k k res res nr res res nr (G.531)F = T F, F = F − F , T F = 0 0 N 0 0 N 0 and for k ≥ 0 −1 nr res nr res (G.532) f ◦ ◦ f ◦ f = ◦ f Y F +F F +F k Y k k k k+1 k+1 where for any k, 2π/q res res F ◦ φ = F . k J∇ r k −δ /2 By Corollary G.3 there exists Y ∈ O (e W ) such that k σ nr nr −1 nr (G.533) []· Y =−T F , Y −δ /2 qδ F . k k k W N k e W k k k res nr Let F := F + F and compute using Proposition 4.7 k k −1 nr res nr res f ◦ ◦ f ◦ f = ◦ f Y F +F F +F +[]·Y +DF O (Y ) −δ /2 1 k Y k k k k k k k k k e W −1 = ◦ f nr res nr res nr R F +T F +F +qδ DF O (F ) N k −δ /2 1 k N k k k k e k W nr res = ◦ f F +F k+1 k+1 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 159 with nr nr −2 nr F = R F + q(ρ δ ) F O (F ) N k k k W 1 k+1 k k k (G.534) res res nr res F = T F + F . k+1 N k k nr res nr res nr res In particular since F = F + F = F + T F + F k+1 k+1 k+1 k+1 N k k nr res nr nr (G.535)F = F + F + T F − F . k+1 k k+1 N k k ∗ ∗ If we deﬁne ε =F , ∗= nr, res, ε =F we get from (G.531)and W k k W k k k k Lemma 5.2 nr −1 (G.536) ε δ ε qε 0 0 0 0 and from (G.534), (G.535) and Lemma 5.2 that for some a > 0 nr −1 −δ N/2 nr −a nr ε δ e ε + qε (ρ δ ) ε k k k ⎨ k+1 k k k res res −1 nr (G.537) ε ε + δ ε k+1 k k −1 nr nr ε = ε + O(ε + δ ε ) k+1 k k+1 k k provided for some a > 0 (that we can assume ≥ 4) −a nr (G.538) (ρ δ ) ε < 1. k k From now on we deﬁne a = 2a + 2, see (G.529), (G.539) ε ≤ ρ with a = 2a + 2 ≥ 10; 0 3 notice that this implies (see (G.536)) nr 2a (G.540) ε ≤ ρ . 0 0 ∗ nr res Let k be the largest integer for which the sequences ε ,ε ,ε are deﬁned. We notice k k −1/3 that for k < min(k ,ρ ) one has from (G.524) −4/9 −1/8 −4/9 −1 −1 −1 −1 −2 (ρ δ ) ρ qρ A ρ ρ ρ ≤ ρ . k k 0 0 0 0 0 0 k −1/3 −4/9 k −1 nr nr Since ε = ε + O(δ ε ) we get that for k+ 1 ≤ ρ , ε = ε + O(ρ ε ) k 0 k 0 k j=0 k 0 0 j=0 k −1/3 −1/8 hence if k < min(k ,ρ ) (recall q ≤ ρ ) 0 0 −a nr −(2a+1) −(2a+2) nr nr qε (ρ δ ) ε ≤ C ρ ε + ρ ε ε . k k k 0 k 0 0 k k j=0 −3/4 −1 −1 −2 −1 −1 On the other hand, from (G.524), q N = q ρ A ≥ A ρ hence, if k + 1 ≤ 0 0 −1/3 −3/4 4/9 −11/36 −1 −1 4/3 −δ N ρ one has δ N = q N/(k + 1) ≥ ρ ρ = ρ and thus δ e ≤ ρ if ρ k 0 0 0 0 0 0 k 160 RAPHAËL KRIKORIAN −1/3 is small enough. The outcome of this is that for k + 1 ≤ ρ one has (we use condition (G.539)) nr −(2a+3) nr nr ε ≤ Cρ 1 + ρ ε ε . k+1 0 k k j=0 Since for ρ 1 one has (cf. (G.540)) nr 2a a (G.541) ε ≤ ρ ≤ (ρ δ ) 0 0 0 0 and we can thus apply Lemma F.1 with α = ρ ,toget −1/3 ∗ nr k nr −k (G.542) k ≥ ρ , ∀0 ≤ k ≤ k ,ε ≤ (2Cρ ) ε ≤ e qε . ∗ 0 0 0 k 0 We now set res res nr nr −1 Wh F = F , F = F , g = f ◦···◦ f ∗ ∗ Wh k k Y ∗ k −1 Wh 2 −δ /2 where Y is a C Whitney extension of (Y , e W ) given by Lemma 2.2. The con- j j jugation relation (G.525) then holds and the conclusion (G.528) is satisﬁed since from (G.542) −1/3 −2/q nr −ρ e W ⊂ W ∗ , F −2/q e qε . h,D(c,ρ) k e W 0 h,D(c,ρ) To check (G.527) we just notice that from (G.533) 2 −4 g − id 1 q ρ ε . C h 0 res res res ∗ nr res nr Finally, since F = T F + T ( F ) and T F = 0(cf. (G.531)) one has from the N N k=0 k N 0 nr k nr −(k−1) inequality ε ≤ (2Cρ ) ε ρ e ε (ρ 1, k ≥ 1) 0 0 0 0 k 0 k k k ∗ ∗ res res res nr nr nr F = T F + T ( F ), F ≤ ε qρ ε W ∗ 0 0 N N k k k k k=1 k=1 k=1 which gives conclusion (G.526): res res F = T F + O(qρε ) . G.3 Proof of Proposition G.1.— We apply Proposition G.4 and we write using Lemma 4.6 ◦ f nr res = ◦ ◦ f res ◦ f nr res nr F +F 2π(p/q)(r−c) −2π(p/q)(r−c) F F +DF O (F ) W 1 h,U = ◦ ◦ f res ◦ f cor 2π(p/q)(r−c) −2π(p/q)(r−c) F F ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 161 with cor a −a nr (G.543) F −1/q q ρ F −2/q e W e W h,D(c,ρ) h,D(c,ρ) provided for some a > 0 a −a nr q ρ F −2/q < 1. e W h,D(0,ρ) The inequalities (G.524)and (G.528) show that this last condition is satisﬁed if ρ 1. We now observe that res res f = ◦ f F M (F) F −M (F) 0 0 and that ◦ = . −2π(p/q)(r−c) M (F) −2π(p/q)(r−c)+M (F) 0 0 If we set g = g RNF res res = − 2π(p/q)(r − c) + M (F) and F = F − M (F) 0 0 and recall (G.525) we ﬁnd the conjugation relation (G.514). −1/(10q) Note that sincewehaveassumedthat F ∈ O (e W ) satisﬁes (G.529)we σ h,D(c,ρ) −1/q have ∈ O (e W ) ∩ TC(2A, 2B) and the ﬁrst inequality of (G.515)issatisﬁed. σ h,D(c,ρ) The other inequalities of (G.515) are consequences of (G.526)and (G.527)and (G.516)is a consequence of (G.528), (G.524)and (G.543). Remark G.2. — Notice that from the ﬁrst inequality of (5.111) in Lemma 5.2 res (G.544) F F . W W −1/q −1/q h,D(c,ρ) e h,D(c,e ρ) Appendix H: Approximations by vector ﬁelds The main result of this Section is the following proposition on the approximation of an exact symplectic diffeomorphism close to an integrable one by a vector ﬁeld. Proposition H.1. — There exists a constant C > 0 for which the following holds. Let 0 <ρ < 1, F ∈ O (T × D(0,ρ)) and ∈ O (D(0,ρ)), (r) = O(r ).If ρ> 0 is small σ h σ 1/3 enough, h ρ and −9 (H.545) C × (ρh) F < 1 h,ρ 162 RAPHAËL KRIKORIAN then, there exist ∈ O (D(0,ρ/2)), A (F) ∈ O (T × D(0,ρ/2)) such that σ 3 σ h/2 (H.546) ◦ f = ◦ f F A (F) with 1/4 (H.547) = + F ◦ + O(ρ F ) −/2 h,ρ 1/4 (H.548) = + F + O(ρ F ) h,ρ and −1/4 (H.549) A (F) < exp(−ρ )F . 3 h/2,ρ/2 h,ρ The proof of this proposition is given in Section H.2. H.1 Auxiliary result. Proposition H.2. — Let ρ> 0, (r) = O(r ), ∈ O (D(0,ρ)), F, G ∈ O (T × σ σ h D(0,ρ)) such that (C some universal constant) −4 (H.550)C × (ρδ) (F +G )< 1. h,ρ h,ρ 1/3 −δ Then for any h/2 >δ ρ , there exists A(F, G) ∈ O (e (T × D(0, ρ))) such that σ h (H.551) = ◦ ◦ f +F+G +F G◦ A(F,G) /2 with −4 −3 −δ (H.552) A(F, G) (ρδ) (F +G ) + ρδ G . h−δ/2,e ρ h,ρ h,ρ h,ρ Proof. — To simplify the notations we denote W = W and we assume that h,D(0,ρ) ω(r) := ∇(r), ω(0) = 0satisﬁes ω(r) = r + O(r ). If −2 (δρ) max(F ,G )< 1 h,ρ h,ρ −2δ t t t the images of the domain e Wby the ﬂows , , ,0 ≤ t ≤ 1, are contained +F +F+G −δ in e W. Let us denote σ := max(DF ,DG ) h,δ h,δ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 163 −2δ and for x = (θ , r) ∈ e Wand t ∈[−1, 1] t t (t, x) = (x) − (x). +F+G +F By classical theorems on ODE’s for t ∈[−1, 1] t t (t,·) = O(σ ), − = O(σ ). +F On the other hand one has t t (H.553) (t, x) = J∇( + F + G) ◦ (x) − J∇( + F) ◦ (x) +F+G +F dt = (I)(t, x) + (II)(t, x) + (III)(t, x) with t t (I)(t, x) = J∇ ◦ (x) − J∇ ◦ (x) +F+G +F t t (II)(t, x) = J∇ F ◦ (x) − J∇ F ◦ (x) +F+G +F (III)(t, x) = J∇ G ◦ (x). +F+G t t Since (x) = (x)+ ( (t, x), (t, x)) and − = O(σ ) one has (note θ r +F +F+G +F that r ◦ = r + O(σ )) +F t t ω(r ◦ (x) + (t, x)) − ω(r ◦ (x)) +F +F (H.554) (I)(t, x) = ∂ω(r) (t, x) + O(σ| (t, x)|) r r and (H.555) |(II)(t, x)|= O(|D F||(t, x)|). We have using the fact that − = O(σ ) and ω(r) = O(r) +F+G (H.556) (III)(t, x) = J∇ G(θ + tω(r), r) + O(εD G) 2 2 ∂ G(θ , r) + tω(r)∂ G(θ , r) + O(ρ ∂ ∂ G) r r θ r θ 2 3 −∂ G(θ , r) − tω(r)∂ G(θ , r) + O(ρ ∂ G) θ θ + O(σD G). 164 RAPHAËL KRIKORIAN Summing (H.554), (H.555), (H.556) and integrating (H.553)gives (t, x) ∂ω(r) (s, x)ds θ r (H.557) = (t, x) 2 2 t∂ G(θ , r) + (t /2)ω (r)∂ G(θ , r) θ r 2 2 −t∂ G(θ , r) − (t /2)ω (r)∂ G(θ , r) + O(ε +|D F|) |(s, x)|ds + A with 2 2 2 A = O(σD G) + O(ρ D∂ G). Lemma H.3. — One has 2 2 2 |(t, x)|≤ A := O(DG+ ρD∂ G)+ O(σD G)+ O(ρ D∂ G). 2 θ Proof. — From (H.557) and the fact that ∂ω(r) 1 |(t, x)|≤ C(1+ε+D F ) |(s, x)|ds+ O(DG+ ρD∂ G)+ A h,ρ θ 1 and we conclude by Grönwall inequality. Looking at the second component of (H.557)gives (ω(r) = O(r)) 2 2 (t, x)=−t∂ G(θ , r) + O(ρ∂ G) + O((ε +D F)A ) + A r θ 2 1 hence (integrating again and putting the result in (H.557)) (t, x) (H.558) (t, x) 2 2 −∂ω(r)(t /2)∂ G(θ , r) + t∂ G(θ , r) + (t /2)ω (r)∂ G(θ , r) θ r θ r −t∂ G(θ , r) − (t /2)ω (r)∂ G(θ , r) + O(A ) with 2 2 A = O(ρ∂ G) + O((ε +D F)A ) + A . 3 2 1 Taking t = 1gives (x) +F+G = (x) +F ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 165 −(∂ ω (r)/2)∂ G(θ , r) + ∂ G(θ , r) + (ω (r)/2)∂ G(θ , r) θ r θ r −∂ G(θ , r) − (ω (r)/2)∂ G(θ , r) + O(A ). On the other hand ∂ (G(θ − ω(r)/2, r)) −∂ (G(θ − ω(r)/2, r)) −(∂ ω (r)/2)∂ G(θ − ω(r)/2, r) + ∂ G(θ − ω(r)/2, r) θ r −∂ G(θ − ω(r)/2, r) hence J∇(G ◦ ) ◦ −/2 −(∂ ω (r)/2)∂ G(θ + ω(r)/2, r) + ∂ G(θ + ω(r)/2, r) θ r −∂ G(θ + ω(r)/2, r) and from Taylor Formula and the fact that ω(r) = O(r) (x) − (x) = J∇(G ◦ ) ◦ + O(ρ∂ G) + O(A ). +F+G +F −/2 3 Since = + O(σ ), this means that +F = (id + J∇(G ◦ ) ◦ + O(σD G) + O(A ) +F+G −/2 +F 3 thus = ◦ + O(A ) +F+G G◦ +F 3 −/2 or = f ◦ ◦ +F+G O(A ) G◦ +F 3 −/2 with −2 2 −1 −2 −2 A ((ρδ) σ + ρ (ρδ) δ + (σ + (ρδ) σ) −1 −1 −1 −2 × ((ρδ) + ρ(ρδ) δ ) + ρδ )G h,ρ −3 −3 ((ρδ) σ + ρδ )G h,ρ −4 −3 (ρδ) (F +G ) + ρδ G h,ρ h,ρ h,ρ provided −4 −3 (ρδ) (F +G )< 1,ρδ < 1. h,ρ h,δ 166 RAPHAËL KRIKORIAN To conclude we observe that if we apply the preceding formula with − and −F instead of ,F = f ◦ ◦ −−F−G O(A ) −G◦ −−F 3 /2 and inverting = ◦ + f . +F+G +F G◦ O(A ) /2 3 Corollary H.4. — Under the same conditions of Proposition H.4 one has ◦ f = ◦ f +F G +F+G◦ A (F,G) −/2 2 with −4 −3 −δ (H.559) A (F, G) (ρδ) (F +G ) + ρδ G . 2 h−δ/2,e ρ h,ρ h,ρ h,ρ Proof. — If we apply (H.551)with G ◦ instead of G we get −/2 = ◦ ◦ f +F+G◦ +F G A(F,G◦ ) −/2 −/2 hence −1 −1 ◦ f = ◦ f ◦ ◦ f +F G +F+G◦ G −/2 A(F,G◦ ) G −/2 = ◦ f +F+G◦ A (F,G) −/2 2 where A (F, G) = A(F, G ◦ ) + O(|DG||D G|) satisﬁes (H.559)(cf. (H.552)). 2 −/2 3/2 H.2 Proof of Proposition H.1.— Let δ = c/(k + 1) , h = h − δ /2, ρ = (3/4)ρ , k k k 0 −δ ρ = e ρ and c chosen such that h ≥ h/2, ρ ≥ ρ/2for all k ∈ N. Using Corollary H.4 k k k we construct sequences S , G such that S = 0, G = F k k 0 0 (H.560) ◦ f = ◦ f +S G +S G k k k+1 k+1 S = S + G ◦ k+1 k k −/2 (H.561) G = A (S , G ) k+1 2 k k with S ≤S +G k+1 h ,ρ k h ,ρ k h ,ρ k+1 k+1 k k k k and −3 −4 (H.562) G ρ δ G + (ρ δ ) (S +G )G k+1 h ,ρ k k h ,ρ k k k h ,ρ k h ,ρ k h ,ρ k+1 k+1 k k k k k k k k k ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 167 as long as −4 (ρ δ ) (S +G )< 1. k k k h ,ρ k h ,ρ k k k k With ε =G and σ := S we have (s = 0) n n h ,ρ n n h ,ρ 0 n n n n −3 −4 (H.563) ε ≤ C(ρ δ + (ρ δ ) ε )ε k+1 k k k j k j=0 (H.564) σ ≤ σ + O(ε ) k+1 k k −4 as long as (ρ δ ) (s + σ )< 1. k k k k Let k be the largest integer for which these sequences are deﬁned. We observe 1−3/4 −3 ∗ 1/4 −1 −2 1/4 that for k < min(k ,ρ ) one has (ρ δ ) ≤ ρ and ρ δ ≤ ρ = ρ ;hence,if k k k k k ∗ −1/4 k = min(k ,ρ ) one has 1/4 −9 ∀ k < k,ε ≤ Cρ (1 + ρ ε )ε . k+1 j k j=0 We are in position to apply Lemma F.1 with α = ρ , θ = 1/4, a = 9: since condition (F.483)issatisﬁed(cf. (H.545)) one has ∗ 1/4 ∗ k/4 k ≥ ρ , ∀ k ≤ k ,ε ≤ (2Cρ) ε k 0 and also 1/4 s − ε ≤ ε ρ ε . k 0 j 0 j=1 To conclude the proof we set = S , A (F) = F . k k Appendix I: Adapted KAM domains: lemmas I.1 Proof of Lemma 10.1. I.1.1 Proof of the RHS inequality of (10.306). — From (7.193) and the deﬁnition (10.300)of i (ρ),for every (k, l) ∈ E ,0 < k < N ,0≤|l|≤ N one has − i (ρ)−1 i (ρ)−1 i (ρ)−1 − − − (i (ρ)−1) − −1 D(c , K ) ∩ D(0, 2ρ)=∅, l/k i (ρ)−1 (i (ρ)−1) (i (ρ)−1) − − hence |c | >ρ . Since ω (c ) = l/k,wededucefromthe fact that l/k i (ρ)−1 l/k (i (ρ)−1) satisﬁes a (2A, 2B)-twist condition (7.166)that |(l/k) − ω |=|ω(c ) − ω(0)|≥ l/k 168 RAPHAËL KRIKORIAN −1 (2A) ρ . By Dirichlet Approximation Theorem, for any L ≥ 1 there exist k, l ∈ Z,0 < |k|≤ Lsuch that |ω − (l/k)|≤ ,hence |k|L −1 (I.565) (2A) ρ ≤ . |k|L 1 1 However, since ω ∈ DC(τ ) one has |ω − (l/k)|≤ hence 0 1+τ 0 |k| |k|L 1/τ L |k|. This and (I.565)yield −(1+1/τ ) ρ L . In particular if one chooses L = N − 1 N one gets i (ρ)−1 i (ρ) − − −(1+1/τ ) ρ N i (ρ) which proves the inequality of the RHS of (10.306). I.1.2 Proof of the LHS inequality of (10.306). — Let us prove the second inequality of (10.306). By deﬁnition of i (ρ) there exists (l, k) ∈ Z ,0 < k < N , |l|≤ N such − i (ρ) i (ρ) − − that (cf. (7.193)) (i (ρ)) − −1 D(c , 2K ) ∩ D(0, 2ρ) = ∅. l/k i (ρ) (i (ρ)) − 1+τ In particular (cf. (7.163)) |c |≤ 3ρ . Since ω ∈ DC(κ, τ ), |ω(0)− (k/l)|≥ κ/k and l/k (i (ρ)) − 1/2 1+τ from (7.192) |ω (0) − ω (c )|≥ κ/k − 2ε ; by the twist condition 6Aρ ≥ i (ρ) i (ρ) l/k − − (i (ρ)) − −(1+τ) 2A|c |≥ κ N − ρ hence l/k i (ρ) −(1+τ) ρ N i (ρ) which shows that the LHS of (10.306) holds. Estimates (10.304), (10.305) are then immediate. I.2 Proof of Items 1, 2, 4 of Proposition 10.2.— Recall that from (10.307) ((3/2)ρ) i−1 (j) −1 U = D(0,(3/2)ρ) D(c , s K ), j,i−1 i j=1 (k,l)∈E l/k j (I.566) i−1 m=j s = e ∈[1, 2] j,i−1 (j) where E ⊂{(k, l) ∈ Z , 0 < k < N , 0 < |l|≤ N }, ω (c ) = l/k. In particular, any j j j j l/k (j) −1 D ∈ D(U ) is of the form D = D(c , s K ),where j ≤ i − 1, (k, l) ∈ E . i j,i−1 j l/k j Lemma I.1. — If D ∈ D (U ) then j ≥ i (ρ). (3/2)ρ i − ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 169 Proof. — Since D(0, 2ρ) = D(0, 2ρ) ∩ U ,from(I.566)for all j ≤ i (ρ) − 1, i (ρ) − (j) −1 (k, l) ∈ E one has |c |≥ 2ρ + K . On the other hand, if D ∈ D(U ) is of the form j i l/k j (j) −1 D = D(c , s K ),where j ≤ i − 1, (k, l) ∈ E and intersects D(0,(3/2)ρ) one has j,i−1 j l/k j (j) −1 −1 |c |≤ (3/2)ρ + 2K < 2ρ + K hence j ≥ i (ρ). l/k j j From (7.193) and Lemma I.1 we can thus write (j) ((3/2)ρ) i−1 −1 U = D(0,(3/2)ρ) D(c , s K ), j,i−1 i l/k j j=i (ρ) (k,l)∈E − j (I.567) i−1 m=j s = e ∈[1, 2]. j,i−1 We deﬁne i−1 Q = {l/k,(k, l) ∈ E } i j j=i (ρ) and for t ∈ Q j(t, i) = min{j : j ∈ N∩[i (ρ), i − 1],(k, l) ∈ E and l/k = t} − j (j(t,i)) c(t, i) = c , s(t, i) = s . j(t,i),i−1 Deﬁne for i (ρ) ≤ j < i ≤ i (ρ), − + −1 κ = s K . j,i j,i−1 We observe that from the inequality N ≤ N ,for any i (ρ) ≤ j < i ≤ i (ρ), i (ρ) − + + i (ρ) i (ρ) ≤ j < i ≤ i (ρ), (i, j) = (i , j ), one has − + −2 1/2 κ + κ N , ε |κ − κ |; j,i j ,i j,i j ,i max(j,j ) min(j,j ) if j < j or j = j and i < i , one has κ <κ hence from Lemma 7.3,Item(2)for j,i j ,i (k, l) ∈ E , (k , l ) ∈ E one has j j (j) (j ) either l/k = l /k and D(c ,κ ) ∩ D(c ,κ )=∅ j,i j ,i l/k l /k (I.568) (j) (j ) or l/k = l /k and D(c ,κ ) ⊂ D(c ,κ ). j,i j ,i l/k l /k As a consequence, for j ∈ N ∩[i (ρ), i − 1], (k, l) ∈ E , t = l/k, one has the inclusion − j (j) −1 −1 D(c , s K ) ⊂ D(c(t, i), s K ), and therefore (cf. (I.567), (10.308)) j,i−1 j(t,i),i−1 l/k j j(t,i) ((3/2)ρ) −1 U = D(0,(3/2)ρ) D(c(t, i), s K ). j(t,i),i−1 i j(t,i) t∈Q 1/2 46 −1 This is clear if j = j ;if j = j observe that if i = i , ε |s − s |K . j,i−1 j,i −1 i (ρ) j − 170 RAPHAËL KRIKORIAN This implies that any D ∈ D (U ) is of the form (3/2)ρ i −1 (I.569)D = D(c(t, i), s K ), t ∈ Q , j(t, i) ≤ i − 1. j(t,i),i−1 i j(t,i) Proof of item 1 of Proposition 10.2.— This is a consequence of (I.569)and (I.568). Proof of item 2 of Proposition 10.2.— One can write for some t ∈ Q , t ∈ Q ,D = i i −1 −1 D(c(t, i), s K ),D = D(c(t , i ), s K ) and from Lemma 7.3,Item(2)if j(t,i),i−1 j(t ,i ),i −1 j(t,i) j(t ,i ) D ∩ D = ∅ one has t = t . On the other hand since t = t ∈ Q ⊂ Q one has j(t, i ) = i i j(t, i).Wenow usethe fact that s ≤ s . j(t,i ),i −1 j(t,i),i−1 Proof of item 4 of Proposition 10.2.— Let us prove that D ∈ D (U ) is a subset of ρ i (ρ) U . If this were not the case, there would exist D ∈ D(U ) such that D ∩ D = ∅;in i i D D particular D ∈ D (U ) andfromitem 2 D ⊂ D; but this contradicts the deﬁnition (3/2)ρ i of i .Hence D ⊂ U . D i This latter inclusion and (I.567) applied with i = i show that one has D ∩ (j) −1 D(c , s K ) =∅ for all i (ρ) ≤ j ≤ i − 1, (k, l) ∈ E . As a consequence D = j,i−1 − D j l/k j (j) −1 D(c , s K ) for some j ≥ i , (k, l) ∈ E . j,i (ρ)−1 D j l/k + j On the other hand, by deﬁnition of i there exists D ∈ D (U ) of the form D ρ i +1 −1 D = D(c , s K ) with j ≤ i , s ∈[1, 2] (cf. (I.569)) such that D ⊂ D. One hence have −1 −1 s K ≥ K thus j ≥ i . We conclude that j = i . j,i (ρ)−1 D D + j i Appendix J: Classical KAM measure estimates J.1 A lemma. — If A and B are two sets we denote by A"B = (A ∪ B) (A ∩ B) their symmetric difference. Lemma J.1. — Let A = I I ,where I ⊂ R is an interval and all the intervals I are j j j∈J 1/2 1 disjoint. Then if |I | ≤ 1 and if g : M → M is a C -symplectic diffeomorphism such that j R R j∈J 1/2 g − id 1 ≤ 1/10, then one has Leb(W " W ) g − id . C A g(A) 0 Proof. — We can assume that the intervals I are contained in I. Recall that 1 =|1 − 1 | and notice that since the intervals I are pairwise disjoint one has A " B A B j 1 = 1 − 1 hence W W W A I I j∈J 1 = χ − χ W " g(W ) j A A j∈J ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 171 where χ = 1 − 1 , χ = 1 − 1 . This gives W g(W ) j W g(W ) I I I I j j Leb (W " g(W )) M A A =χ − χ 1 j L j∈J ≤χ 1 + χ 1 L j L j∈J ≤ Leb (W " g(W )) + Leb (W " g(W )). M I I M I I R R j j j∈J On the other hand, if I is an interval, there exist intervals I ⊂ I ⊂ Isuch thatW ⊂ −1 g(W ) ⊂ W and max(|I " I|,|I " I||) ≤ 2max(g − id 0 ,g − id 0 ) ≤ Cg − id 0 , I I C C C C > 0 depending only on M (recall that we have assumed g − id 1 is small enough). This is clear in the (AA)-case and in the (CC) or (CC*)-case it follows from the (AA)-case using the symplectic changes of coordinates ψ and ϕ (4.79), (4.77). Therefore, since g is symplectic, Leb (W " g(W )) ≤ Cmin(g − id 0 , Leb (W )). M I I C M I R j j R j 1/2 1/2 In particular Leb (W " g(W )) ≤ Cg− id Leb (W ) and since Leb (W ) ≤ M I I 0 M I M I R j j R j R j |I | the conclusion follows. J.2 Proof of Theorem 12.1.— We use the notations of Section 7 and Propositions 7.1, 7.2, 7.5 and Remark 7.1. We apply Proposition 7.5–Remark 7.1 with m = 1 and Proposition 4.3 with A = −2δ e U, L = L , A = U = U, 1,Prop. 7.5 1 Leb (W −2δ L(f , W )) M 1 R R∩e U R∩U 1/2 −2δ ≤ C × (Leb (R ∩ (e U L))+g − id ) R 1,∞ 0 1 1 1/8 2(a +3) 2(a +3) 0 0 ε + ε ε . Appendix K: From (CC) to (AA) coordinates Sometimes we need to reduce the (CC)-case to the (AA)-case, for example when deﬁning the Hamilton-Jacobi Normal Form in Section 8 or in Section 16. For α ∈]0,π[ deﬁne the angular sector (ρ) ={r ∈ D(0,ρ), arg(r)/∈[−α, α]} − + and (ρ)=− (ρ). Recall the deﬁnition of the maps ψ , cf. (4.79)ofSection 4.1. α α 172 RAPHAËL KRIKORIAN CC CC h/2 2 CC Lemma K.1. — Let c ∈ R, F ∈ O (W ), ε = Ce D F CC , h,D(c,2ρ) W h,D(c,2ρ) −2 −2 (K.570)Cδ ρ ε< 1 and α ∈]δ, π − δ[. CC 3 AA (1) If c = 0 and F = O (z,w) there exists F ∈ O (T × (ρ − 4δ)) such that on σ h−δ ± α+4δ T × (ρ − 4δ) one has h−4δ α+4δ −1 AA CC CC AA CC (K.571) f = ψ ◦ f ◦ ψ , F = F ◦ ψ + O (F ). ± ± 2 F F ± ± AA (2) If |c| > 4ρ , there exists F ∈ O (T × D(0,ρ)) such that (K.571) holds. σ h−δ Proof. — We prove item (1), the proof of item (2)isdoneinasimilar(andeven h−δ simpler) way. From ε ≤ δ and f (0) = 0we get that if z,w satisfy |z|,|w| < e (ρ − 1/2 ± 3δ) , r =−izw ∈ (ρ− 3δ) then z, w deﬁned as ( z, w) = f CC (z,w) satisfy | z− z|≤ α+3δ −h/2 −h/2 ε h 1/2 ε h 1/2 e ε|z|, |w − w|≤ e ε|w| and thus | z|≤ e |z|≤ e ρ and |w |≤ e |w|≤ e ρ ;on the other hand if r =−i zw one has z/w (K.572) | r − r|≤ 3ε|r|, | arg( r) − arg(r)|≤ 3ε, − 1 ≤ (5/2)ε. z/w Since ε< δ, ± ± CC f ◦ ψ (T × (0,ρ − 3δ)) ⊂ ψ (T × (0,ρ)) ± h−3δ ± h α+3δ α AA −1 ± hence f := ψ ◦ f CC ◦ ψ : T × (0,ρ − 3δ) → T × (0,ρ) is well deﬁned. F ± h−3δ h ± α+3δ α −1 AA On the other hand if ψ (z,w) = (θ , r), f (θ , r) = (θ, r), ψ (θ, r) = ( z, w) one has ± ± from (K.572) and Lemma M.1 max(|θ − θ| ,| r − r|) ≤ 3ε 2π Z hence AA f − id ≤ 3ε T × (0,ρ−3δ) h−3δ α+3δ AA and from Remark 4.2, Lemmata 4.4, 4.5 and condition (K.570) there exists F ∈ ± AA AA O(T × (0,ρ − 4δ)) such that f = f and h−4δ α+4δ AA AA (K.573) f = φ ◦ f . AA F O (F ) ± J∇ F ± To get the second estimate in (K.571)wenoticethat CC CC f = φ ◦ f CC F O (F ) J∇ F 2 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 173 hence −1 1 f AA = ψ ◦ φ ◦ f CC ◦ ψ CC F O (F ) ± ± J∇ F 2 1 −1 = φ ◦ ψ ◦ f CC ◦ ψ CC O (F ) ± J∇(F ◦ψ ) ± 2 = φ ◦ f CC CC O (F ) J∇(F ◦ψ ) 2 and from (K.573) AA CC CC F = F ◦ ψ + O (F ). ± 2 CC CC Remark K.1. —If f = ◦ f CC we have (cf. Section 4.2). −1 CC AA AA ψ ◦ f ◦ ψ = ◦ f . Appendix L: Lemmas for Hamilton-Jacobi Normal Forms L.1 Proof of Lemma 8.3.— Since ∂ (r) 1(cf. (8.218)), (8.222)and (8.220)show that there exists e : T → C, e ∈ O (T ) such that 0 qh/3 0 σ qh/3 −1 2 (L.574) ∀ θ ∈ T ,∂ (θ , e (θ )) = 0, e (qρ) q ε. qh/3 r 0 0 T qh/3 We now make a Taylor expansion: using (L.574) we see that (L.575) (θ , r) = (θ , e (θ ) + (r − e (θ )) 0 0 2 2 = (θ , e (θ )) + (1/2)∂ (θ , e (θ ))(r − e (θ )) 0 0 0 3 k k−3 + (r − e (θ )) ∂ (θ , e (θ ))(r − e (θ )) 0 0 0 k! k=3 and if we deﬁne (L.576) (θ) = (1/2)∂ (θ , e (θ )), e (θ )=−(θ , e (θ ))/ (θ ) 0 1 0 one gets for some p(θ , r) 2 3 (θ , r) = (θ) −e (θ ) + (r − e (θ )) + (r − e (θ )) p(θ , r) 1 0 0 = (θ , r − e (θ )) −1 −2/q with ∈ O(T × D(0, e qρ/2 − Cqρ ε)) ⊂ O(T × D(0,ρ )) qh/3 qh/3 q 2 3 (θ , r) = (θ) r − e (θ ) + r p(θ , r + e (θ )) ; 1 0 this gives the desired form for (θ , r) if one sets f (θ , r) = p(θ , r + e (θ )). 0 174 RAPHAËL KRIKORIAN The estimates (8.228)on e , e , are then clear from (L.574), (L.576). Let us check 0 1 the one on f .From(8.223)and (8.227)wehave 2 3 (θ)(r − e (θ ) + r f (θ , r)) 2 i 3 =: r + f (θ )(r + e (θ )) + r (b(r) + f (θ , r + e (θ ))) i 0 0 i=0 hence from (8.225) and the ﬁrst two inequalities of (8.228) 3 −1 −3 2 r f (θ , r) − (θ) b(r) − f (θ , r + e (θ )) (qρ) q ε and by the maximum principle −1 sup f (θ , r) − (θ) b(r) − f (θ , r + e (θ ))) (θ ,r)∈T ×D(0,ρ ) qh/3 q −3 −3 2 ρ (qρ) q ε 1. We the conclude by (8.218)and (8.225). L.2 Square roots. ∗ 2 1/2 Lemma L.1. — Let a ∈ C . There exists a unique function m (z) = z(1+ a/z ) univalent 1/2 on C D(0,|a| ) such that 2 2 −1 (L.577) m (z) = z + a, m (z) = z + O(z ). 1/2 It satisﬁes for z, z ∈ E := {w ∈ C,|w| > L|a| } (L > 3) 2 m (z) − m (z ) 2 a a −2/L 1/L (L.578) (2/π )e ≤ ≤ (π/2)e z − z 2 1/2 Proof. — The existence and uniqueness of m (z) = z(1 + (a/z )) is clear. Note that the inverse for the composition of m is m and that if L > 2 m (E ) ⊂ a −a a L 2 −1/2 E . On the other hand the derivative of m (z) is equal to ∂ m (z) = (1 + a/z ) and 3L/4 a z a −1/2 1/L since for t ∈[0, 1/2], (1 − t) ≤ 1 + t onegetsfor z ∈ E (L > 2) |∂ m (z)|≤ e . L z a Now any two points z, z ∈ E can be joined by a path in E the length of which is L L 1/L ≤ (π/2)|z − z |;thusfor any z, z ∈ E , |m (z) − m (z )|≤ (π/2)e |z − z | which is L a a the right hand side inequality of (L.578). To get the left hand side we use the fact that 1/(3L/4) |m (m (z)) − m (m (z ))|≤ (π/2)e |m (z) − m (z )| if L > 3(3L/4 > 2). −a a −a a a a ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 175 2 1/2 L.3 Proof of Lemma 8.5.— From Lemma L.1 z → (z + a) is well deﬁned on 1/2 C {|z| > |a| }. −1/2 Let 0 ≤ s ≤ h/3. We are looking for g(θ , z) = (θ) z(1 + g ˚(θ , z)) such that 2 2 −1 2 z = (θ) z (θ) (1 + g ˚(θ , z)) − e (θ ) 3 −3/2 3 + z (θ) (1 + g ˚(θ , z)) f (θ , g(θ , z)) which can be written as a Fixed Point problem 1/2 e (θ ) −1/2 3 (L.579) g ˚(θ , z)= 1 + (θ) − z(θ) (1 + g ˚(θ , z)) f (θ , g(θ , z)) − 1. Using the estimate on f given by (8.228) one can see that the map " : g ˚ → R.H.S. of −2 (L.579)deﬁnesa 2ρ -contracting map on the ball B(0, CL ) of center 0 and radius −2 −1 CL of the Banach space (O(T × A(λ ,ρ )),· ) provided L and ρ are small sq s,L q ∞ q enough. By the Contraction Mapping Theorem it has a unique ﬁxed point g ˚ in this ball. In other words (L.580) (θ , g(θ , z)) = z . The fact that g ∈ O(T × A(λ ,ρ )) is uniquely deﬁned shows that the various g found qs s,L q for different values of s must agree. Hence g is deﬁned on (T × A(λ ,ρ )). qs s,L q 0≤s≤1 −1 L.4 Proof of Lemma 8.6.— We look for H under the form H(z) = γ z(1 + H(z)). Equation (8.238) can be written as a Fixed Point problem: −1 ˚ ˚ (γ z(1 + H(z)))) (L.581) H(z)=− . −1 ˚ ˚ (1 + (γ z(1 + H(z)))) By Cauchy’s estimates for z ∈ A(λ ,ρ ) s,L q −2 |∂(z)|≤ L . dist(z,∂ A(λ ,ρ )) s,L q −1 −2 Hence if z ∈ A(2λ ,(1/2)ρ ) the map u → (γ zu) is 4L -Lipschitz on {(3/4) ≤ s,L q −2 |u|≤ 4/3} and the map " deﬁned by the R.H.S. of (L.581)is 4L contracting on the −2 ball {H ≤ 2L }. It admits thus a unique ﬁxed point in this ball. A(2λ ,(1/2)ρ ) s,L q Appendix M: Some other lemmas Lemma M.1. — Let z ∈ C 176 RAPHAËL KRIKORIAN 1) One has iz |e − 1|≥ min(1, min|z − 2π l|). l∈Z 2) If z ∈ R, iz |e − 1|≥ min|z − 2π l|). l∈Z iz Proof. — 1) Let η := e − 1. We can assume |η| < 1/2. We can thus deﬁne iz := k−1 k iz iz ln(1 + η) = (−1) η /k such that e = 1 + η = e . There thus exists l ∈ Z such k∈N that z = z − 2π l.But |z |=| ln(1 + η)|≤ 2|η|. 0 0 2) Just use the fact that for |w|≤ π , |2sin(w/2)|≥ (2/π )|w|. Lemma M.2. — Let f ∈ C (T) be such that for some δ ∈]0, 1[, μ> 0 2 0 (M.582) f ≤ δf + μ. L (T) C (T) Then, for some C > 0, −1 −h/(12δ ) (M.583) f 0 ≤ δ μ + e f . C (T) h Proof. — If iikθ (M.584) f (θ ) = f (k)e k∈Z is the Fourier expansion of f , one has for some C > 0and anyN ∈ N −hN (M.585) f ≤ |f (k)|+ e f C (T) h |k|≤N 1/2 −hN (M.586) ≤ (2N + 1) f + e f L (T) 1/2 −hN (M.587) ≤ (3N) (δf 0 + μ) + e f . C (T) h −2 1/2 If we choose N = δ /12 we have (3N) δ ≤ 1/2and −1 −h/(12δ ) (M.588) f ≤ δ μ + e f . C (T) h h ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 177 Appendix N: Stable and unstable Manifolds N.1 The Stable Manifold Theorem. — Let (E,·) be a Banach space, M : E → Ean invertible linear continuous map. Let κ, δ > 0. We say that M is (κ, δ)-hyperbolic if there exist κ> 0 and continuous projectors P ,P satisfying id = P + P ,P P = P P = 0, s u E s u s u u s P MP = P MP = 0such that s u u s −1 −κ max(P MP ,(P MP ) ) ≤ e s s u u −1 max(P ,P ) ≤ δ . s u The spaces E := P E, ∗= s, u, are then M-invariant and are the stable and unstable ∗ ∗ spaces of the linear map M. We shall use the notations M = P MP , ∗= s, u. ∗ ∗ ∗ Let B(0,ρ) ⊂ E be the ball of center 0 and radius ρ> 0. Theorem N.1 (Stable/Unstable Manifold Theorem).— Assume that M is (κ, δ)-hyperbolic as above and let F : B(0,ρ) → E be C . Assume that −1 −1 (N.589) F(0)≤ C δκρ , DF 1 ≤ C δκ. C (B(0,ρ)) Then, if C is large enough (but universal) (1) The map x → Mx+ F(x) has a unique hyperbolic ﬁxed point x such that max(P x,P x) ≤ s u (ρ/4) (in particular, it is located in B(0,ρ/2)). (2) The local stable (resp. unstable) manifold s n W (x; M + F) := {y ∈ B(x,ρ/4), ∀ n ≥ 0,(M + F) (y) ∈ B(x,ρ/2)} loc u n (resp. W (x; M+ F) := {y ∈ B(x,ρ/4), ∀ n ≤ 0,(M+ F) (y) ∈ B(x,ρ/2)}) of the point loc xfor M + F is of the form {x + γ (x ), x ∈ E ∩ B(0,ρ/2)} (resp. {x + γ (x ), x ∈ s s,F s s s u u,F u u E ∩ B(0,ρ/2)})where γ : E → E (resp. γ : E → E )isamapofclass C and u s,F s u u,F u s −1 −1 Dγ ≤ CDF (δκ ) (resp. Dγ ≤ CDF (δκ ) ). s,F B(0,ρ) u,F B(0,ρ) −1 (3) If G satisﬁes also (N.589)thenfor ∗= s, u, Dγ − Dγ ≤ CD(F− G) (δκ ) . ∗,F ∗,G B(0,ρ) (4) If F(0) = 0, DF(0) = 0 then x = 0 and T W (0) = E , ∗= s, u. 0 ∗ loc Notice that the Theorem gives the same size for the domains of deﬁnition of γ , s,F γ . u,F N.2 Proof of Lemma 15.4.— Using the deﬁnition of f (θ , r) = (ϕ, R) one can H + ω see that f (θ , r) = (θ , r) if and only if H + ω (N.590) ∇ H (θ , r)+∇ ω(r) = 0 or equivalently 0 = a(0)θ + ∂ b(0)r 0 = ∂ b(0)θ + ( + ∂ a(0))r + ∂ a(0) + ∂ ω(r). r r r r 178 RAPHAËL KRIKORIAN Solving the ﬁrst equation and inserting it into the second yields ∂ b(0) (N.591) θ =− r a(0) ∂ a(0) + ∂ ω(r) r r (N.592) r =− . (∂ b(0)) 2 r + ∂ a(0) − a(0) We observe that, cf. (15.422), 2 2 2 −1 −2 −qh max(|∂ a(0)|,|(∂ b (0)) / a(0)|) q ν ρ e ε </10 r p/q r q p/q 2 −1 −qh 2 |∂ a(0)| q ρ e ε ,∂ ω(r) = O(r ) r p/q r p/q and deduce by a simple ﬁxed point theorem (in dimension 1) that (N.592) has a unique real solution r ∂ a(0); returning to (N.591) and using (cf. (15.423), (15.422)) 0 r 2 −qh 2 −1 −qh (N.593) a(0) = q ν e ε , |∂ b(0)| q ρ e ε q p/q r p/q p/q we conclude that (N.590) has also a unique solution (θ , r ) ∈ D(0,ρ ) 0 0 p/q −1 2 −2 −qh 2 −1 −qh |θ | ν q ρ ε e , |r | q ρ ε e 0 p/q 0 p/q q p/q p/q −8 −8 72 2 −qh −100 and in particular since ρ = max((c /4) , q ) (cf. (15.405)), q e = O(q ), ε ≤ p/q p/q p/q c (cf. (15.406)), ν qρ (cf. (15.409)), a ≥ 10, one has q p/q 3 p/q 5 5 2 (N.594) (θ , r ) ∈ (D(0,ρ ) × D(0,ρ )) ∩ R . 0 0 p/q p/q We now compute Df (θ , r ). Since ω depends only on the r-variable one has, cf. H + ω 0 0 (4.95) of Lemma 4.6, f = ◦ f H + ω ω H Q Q hence 1 ∂ ω(r ) Df (θ , r ) = Df . 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Publisher: Springer Journals
Copyright: Copyright © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2022
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/s10240-022-00130-2
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Abstract

by RAPHAËL KRIKORIAN To the memory of my father Grégoire Krikorian (1934–2018) ABSTRACT It is well known that a real analytic symplectic diffeomorphism of the 2d -dimensional disk (d ≥ 1) admitting the origin as a non-resonant elliptic ﬁxed point can be formally conjugated to its Birkhoff Normal Form, a formal power series deﬁning a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Pérez-Marco’s theorem (Ann. Math. (2) 157:557–574, 2003) is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when d = 1 the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus. CONTENTS 1. Introduction ...................................................... 2 2. Notations, preliminaries ............................................... 26 3. A no-screening criterion on domains with holes .................................. 33 4. Symplectic diffeomorphisms on holed domains .................................. 38 5. Cohomological equations and conjugations ..................................... 46 6. Birkhoff Normal Forms ................................................ 54 7. KAM Normal Forms ................................................. 60 8. Hamilton-Jacobi Normal Form and the Extension Property ............................ 68 9. Comparison Principle for Normal Forms . ..................................... 84 10. Adapted Normal Forms: ω Diophantine . ..................................... 88 11. Adapted Normal Forms: ω Liouvillian (CC case) ................................. 97 12. Estimates on the measure of the set of KAM circles . ............................... 103 13. Convergent BNF implies small holes ........................................ 109 14. Proof of Theorems C, A and A’ ........................................... 111 15. Creating hyperbolic periodic points ......................................... 113 16. Divergent BNF: proof of Theorems E, B and B’ .................................. 122 Acknowledgements ..................................................... 129 Appendix A: Estimates on composition and inversion .................................. 130 Appendix B: Whitney type extensions .......................................... 135 Appendix C: Illustration of the screening effect . .................................... 138 Appendix D: First integrals of integrable ﬂows ...................................... 139 Appendix E: (Formal) Birkhoff Normal Forms . ..................................... 141 Appendix F: Approximate Birkhoff Normal Forms ................................... 147 Appendix G: Resonant Normal Forms .......................................... 155 Appendix H: Approximations by vector ﬁelds . ..................................... 161 Appendix I: Adapted KAM domains: lemmas . ..................................... 167 Appendix J: Classical KAM measure estimates . . .................................... 170 Appendix K: From (CC) to (AA) coordinates ....................................... 171 Appendix L: Lemmas for Hamilton-Jacobi Normal Forms ............................... 173 Appendix M: Some other lemmas . ........................................... 175 Appendix N: Stable and unstable Manifolds ....................................... 177 References ......................................................... 179 This work was supported by a Chaire d’Excellence LABEX MME-DII, the project ANR BEKAM: ANR-15- CE40-0001 and an AAP project from CY Cergy Paris Université. © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2022 https://doi.org/10.1007/s10240-022-00130-2 2 RAPHAËL KRIKORIAN 1. Introduction We consider in this paper real analytic diffeomorphisms deﬁned on an open set of the d d d d 2d -cartesian space R × R or respectively of the 2d -cylinder (or annulus) (R/2π Z) × R (d ≥ 1), which are symplectic with respect to the canonical symplectic forms dx ∧ dy , j j j=1 d d d d (x, y) ∈ R × R , resp. dθ ∧ dr , (θ , r) ∈ (R/2π Z) × R , and leave invariant j j j=1 d d d d d {(0, 0)}∈ R × R ,resp.the torus T := (R/2π Z) ×{0}⊂ (R/2π Z) × R .Weshall d d d assume that the invariant sets {(0, 0)}∈ R × R , resp. (R/2π Z) ×{0},are elliptic equilib- rium sets in the following sense: there exists ω = (ω ,...ω ) ∈ R ,the frequency vector,such 1 d that d d 2 f : (R × R ,(0, 0)) ý, f = Df (0, 0) ◦ (id + O (x, y)) (1.1) ±2π −1ω spec(Df (0, 0))={e , 1 ≤ j ≤ d} and respectively d d 2 (1.2) f : ((R/2π Z) × R , T ) ý, f (θ , r) = (θ + 2πω, r) + (O(r), O(r )). If ω = ω for i = j (stronger non-resonance condition will be made later), the deriva- i j tive Df (0, 0) of f at the ﬁxed point (0, 0) in (1.1) can be symplectically conjugated to a symplectic rotation and we can thus assume Df (0, 0) is a symplectic rotation: for any x = (x ,..., x ), y = (y ,..., y ), x = ( x ,..., x ), y = ( y ,..., y ) one has (i = −1) 1 d 1 d 1 d 1 d 2π iω x + i y = e (x + iy ) j j j j Df (0, 0) · (x, y) = ( x, y) ⇐⇒ ∀ 1 ≤ j ≤ d. We shall refer to situation (1.1)asthe Elliptic ﬁxed point or the Cartesian Coordinates ((CC) for short) case and to situation (1.2)asthe Action-Angle ((AA) for short) case. Important examples of such diffeomorphisms are provided by ﬂows ( ) ,orby t∈R suitable Poincaré sections on some energy level, of Hamiltonian systems ∂ H ∂ H ∂ H ∂ H x˙ = (x, y), y˙ =− (x, y), resp. θ = (θ , r), r˙=− (θ , r) ∂ y ∂ x ∂ r ∂θ d d d d where H : (R × R ,(0, 0)) → R resp. H : ((R/2π Z) × R , T ) → R (d = d or d = d + 1) is real analytic and satisﬁes 2 2 x + y j j (1.3) (CC)-case H(x, y) = 2π ω + O (x, y), j=1 (1.4) (AA)-case H(θ , r) = 2π ω r + O(r ). j j j=1 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 3 If we denote by the time-1 map of a Hamiltonian H and deﬁne the observable 2 2 r : (x, y) → (1/2)(x + y ), resp. r : r → r (1 ≤ j ≤ d ) we can write (1.1), resp. (1.2), as j j j j j d d 2 (1.5) (CC)-case f : (R × R ,(0, 0)) ý, f = + O (x, y) 2πω,r d d 2 (1.6) (AA)-case f : ((R/2π Z) × R , T ) ý, f = + (O(r), O(r )) 0 2πω,r where ω, r= ω r , r = (r ,..., r ). j j 1 d j=1 The representations (1.5), resp. (1.6), give a very rough understanding of the be- havior of the ﬁnite time dynamics of the diffeomorphism f in a neighborhood of the elliptic equilibrium sets {(0, 0)}, resp. T : it is interpolated by the dynamics of which is 0 2πω,r quasi-periodic in the sense that all its orbits are quasi-periodic with frequencies ω ,...,ω . 1 d Improving this approximation is an old and important problem (it was a central theme of research of the astronomers of the XIXth century; see the references of the very in- structive introduction by Pérez-Marco in [36]) that has a solution at least in the (CC)-case (1.5) if the frequency vector ω is non-resonant:any relation k + k ω +···+ k ω = 0with 0 1 1 d d k , k ,..., k ∈ Z implies that k = k =··· = k = 0. Indeed, after using nice changes of 0 1 d 0 1 d coordinates (symplectic transformations) one can interpolate, in small neighborhoods of the origin, the dynamics of f by quasi-periodic ones with much better orders of approxima- tion and for much longer times. There are two remarkable features of this interpolation: the ﬁrst, is that the frequencies of the interpolating quasi-periodic motions now depend on the initial point and do not necessarily coincide with the frequencies at the origin; the second, is that if we push the order of approximation, these frequencies stabilize in a way. This is the content of the famous Birkhoff Normal Form Theorem, formalized by Birkhoff in the 1920’s [5], [4], [49] and which paved the way to the major achievements of the KAM theory (named after Kolmogorov, Arnold and Moser) in the 1960’s, on the existence of (inﬁnite time) quasi-periodic motions for a wide class of diffeomorphisms of the form (1.1), (1.2); see [29], [1], [32](and[34] for ﬁnite time approximations). We now describe in more details the Birkhoff Normal Form Theorem. 1.1. Birkhoff Normal Forms. — From now on, we assume that ω is non-resonant. We begin with the Elliptic ﬁxed point case ((CC)-case). The ﬁrst statement of the Birkhoff Normal Form Theorem is the following. For any N ∈ N , there exist a poly- nomial B ∈ R[r ,..., r ],B (r) = 2πω, r+ O(r ), of total degree N and a symplec- N 1 d N d d tic diffeomorphism Z : (R × R ,(0, 0)) ý (preserving the standard symplectic form dx ∧ dy and tangent to the identity Z = id + O (x, y))suchthat k k N k=1 −1 2N+1 (1.7)Z ◦ f ◦ Z (x, y) = (x, y) + O (x, y). N B N N 1 −α k k For (x, y)ε-close to (0, 0) and n ∈ N not too large n = O(ε ),0 <α < 1 the iterates f (x, y), k ≤ n (f denotes 2−α k the composition f ◦···◦f , k times) stay ε -close to those of the symplectic rotation, (x, y). 2πω,r 4 RAPHAËL KRIKORIAN d d The diffeomorphism : (R × R , 0) ý is a generalized symplectic rotation i∂ B (r) j N x + i y = e (x + iy ) j j j j (1.8) (x, y) = ( x, y) ⇐⇒ ∀ 1 ≤ j ≤ d 2 2 2 2 (recall r = ((1/2)(x + y ), ...,(1/2)(x + y )))and deﬁnes an integrable dynamics in a 1 1 d d strong sense: every orbit of is quasi-periodic and, in addition, the origin is Lyapunov ∗ d stable. Indeed, for each c = (c ,..., c ) ∈ (R ) ,the d -dimensional torus 1 d 2d 2 2 T := {(x, y) ∈ R , ∀ 1 ≤ j ≤ d, r := (1/2)(x + y ) = c } c j j j j is globally invariant by and the restricted dynamics of on the torus T T := B B c N N d d d d R /(2π Z) is conjugated to a translation T θ → θ + 2πω(c) ∈ T with frequency vector −1 ω(c) = (2π) ∇ B (c). The dynamics of is thus completely understood on the whole N B 2 d d phase space R × R . Here comes the second part of the statement. The polynomials B and the com- ponents of Z − id converge as formal power series when N goes to inﬁnity: B → B ∈ N N ∞ R[[r ,..., r ]],Z → Z ∈ R[[x, y]] and, in the set of formal power series R[[x, y]],one 1 d N ∞ has the following formal conjugacy relation −1 (1.9)Z ◦ f ◦ Z (x, y) = (x, y). ∞ B ∞ ∞ The formal power series B is unique if Z is tangent to the identity and is therefore ∞ ∞ invariant by (smooth or formal) conjugations tangent to the identity; it is called the Birkhoff Normal Form (BNF for short) of f and we shall denote it by BNF(f ): BNF(f ) = B (r ,..., r ) ∈ R[[r ,..., r ]]. ∞ 1 d 1 d On the other hand the formal conjugacy Z , which is called the normalization transforma- tion, is not unique (but if properly normalized is unique). The preceding results hold in the Action-Angle case (1.6) but under a Diophantine assumption on ω (this is stronger than mere non-resonance): (1.10) ∀ k ∈ Z {0}, min|k,ω− l|≥ (τ ≥ d). l∈Z |k| The positive numbers τ and κ are called respectively the exponent and the constant of the 3 ∗ Diophantine condition. One can then prove similarly the existence: (a) for any N ∈ N , When c has some zero components, T is a d -dimensional torus, 0 ≤ d ≤ d , and the restricted dynamics of c c c B on T is again conjugate to a translation on a torus. 3 d The set of vectors of R satisfying a Diophantine condition with ﬁxed exponent τ and ﬁxed constant κ has positive Lebesgue measure if τ> d and if κ> 0 is small enough; for each τ> d , the union of these sets on all κ> 0has full Lebesgue measure in R . ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 5 of a polynomial B ∈ R[r ,..., r ],B (r) = 2πω, r+ O(r ) and of a symplectic dif- N 1 d N d d feomorphism Z : ((R/2π Z) × R , T ) ý (preserving the standard symplectic form N 0 dθ ∧ dr ,Z = id + (O(r), O(r )))suchthat k k N k=1 −1 N N+1 (1.11) Z ◦ f ◦ Z (θ , r) = (θ , r) + (O (r), O (r)) N B (1.12) (θ , r) = (θ +∇ B (r), r) B N ( is called an integrable twist); and: (b) of a formal power series B ∈ R[[r ,..., r ]], B ∞ 1 d the Birkhoff Normal Form, and of a formal symplectic transformation Z = id + 2 ω d (O(r), O(r )) in C (T )[[r ,..., r ]] (the set of formal power series with coefﬁcients in 1 d d ω d the set of real analytic functions T → T) such that one has in C (T )[[r ,..., r ]] the 1 d formal conjugation relation −1 (1.13)Z ◦ f ◦ Z (θ , r) = (θ +∇ B (r), r). ∞ ∞ Again we denote BNF(f ) = B (r ,..., r ) ∈ R[[r ,..., r ]]. ∞ 1 d 1 d All the preceding discussion on Birkhoff Normal Forms holds if we only assume f to be smooth. We can summarize this: d d Theorem (Birkhoff).— Any smooth symplectic diffeomorphism f : (R × R ,(0, 0)) ý d d (d ≥ 1)(resp. f : ((R/2π Z) × R , T ) ý) admitting the origin as a non-resonant elliptic ﬁxed point (resp. of the form (1.6) with ω Diophantine) is formally (strongly) integrable: it is conjugated in ∞ d R[[x, y]] (resp. C ((R/2π Z) )[[r]]) to the formal generalized symplectic rotation (resp. the formal integrable twist) . The formal series BNF(f ) is an invariant of formal conjugation. BNF(f ) We refer to [4]and [49] (Section 24) for a proof of the preceding theorem in the case of symplectic diffeomorphisms of the disk admitting a non-resonant elliptic ﬁxed point and to [13] for the case of Hamiltonian systems admitting a Diophantine KAM torus. We shall reformulate (in the real analytic case) this theorem in Section 6, cf. Propo- sitions 6.1–6.2, and shall give a proof of it in Section E of the Appendix where we mainly concentrate on the AA-case. These formal (and approximate) Birkhoff Normal Forms can be deﬁned in the ∂ H ∂ H more classical setting of Hamiltonian ﬂows x˙ = (x, y), y˙ =− (x, y) (or θ = ∂ y ∂ x ∂ H ∂ H t (θ , r), r˙ =− (θ , r)): f and (B = BNF(f )) are then replaced by ( ) and B t∈R ∂ r ∂θ t 5 ( ) in (1.9), (1.13)(we shallthenwrite B = BNF(H) ). t∈R In the Hamiltonian case, there is a weaker notion of integrability, usually called Poisson integrability, which corresponds to the situation where the considered Hamiltonian has a complete system of functionally independent integrals (observables constant under the motion) which commute for the Poisson bracket. Poincaré discovered [37] that, in 4 k In the C category, one can deﬁne B and Z up to some order N depending on k but one cannot deﬁne in N N general BNF(f ). A more classic equivalent formulation is H = B ◦ Z. 6 RAPHAËL KRIKORIAN general, real analytic Hamiltonian ﬂows do not admit other analytic ﬁrst integrals than the Hamiltonian itself and hence that in general no relation like (1.9) can hold with con- verging Z and B .Siegelproved[48] in 1954 (see also [47], [49], [52], [36]) that, ∞ ∞ whatever the ﬁxed non-resonant frequency vector at the origin ω is, the normalizing con- jugation Z cannot in general deﬁne a convergent series. Indeed, the existence of a convergent normalizing transformation yields real analytic Poisson integrability afact (known to Birkhoff [5]) that is not compatible with the richness of a generic dynamics near a non-resonant elliptic equilibrium. Note that the converse statement is true: real analytic Poisson integrability implies the existence of a real analytic normalizing Birkhoff transformation (cf. [25], [28], [56]). As for the Birkhoff Normal Form itself, H. Eliasson formulated the following natu- ral question [11], [10] (see also the references in [36]): Question A (Eliasson).— Are there examples of real analytic symplectic diffeomorphisms or Hamiltonians admitting divergent (i.e. with a null radius of convergence) Birkhoff Normal Form? The preceding question has an easy positive answer in the smooth case (the map f is only assumed to be smooth): indeed, one can choose f to be of the form f = where : (R , 0) → R is smooth with a divergent Taylor series at the origin; since equalities (1.9)(1.13) only depend on the inﬁnite jet J(f ) of f at 0, the special integrable form of f implies BNF(f ) = J(f ) thus BNF(f ) is diverging. The situation is not so clear if f is real analytic. In contrast with the aforementioned generic divergence of the normalizing 9 10 transformation, there seems to be a priori no obvious dynamical obstruction to the divergence of the Birkhoff Normal Form. The ﬁrst breakthrough in connection with Eliasson’s question came from R. Pérez- Marco [36] who proved, in the setting of Hamiltonian systems having a non-resonant elliptic ﬁxed point, the following dichotomy: Theorem (Pérez-Marco [36]).— For any ﬁxed non-resonant frequency vector ω ∈ R , d ≥ 2, one has the following dichotomy: either for all real analytic Hamiltonian H of the form (1.3) BNF(H) converges (deﬁnes a converging analytic series) or there is a “prevalent” set of such H for which BNF(H) diverges. We refer to Section 1.4 for a precise deﬁnition of “prevalent”. A similar dichotomy holds in the setting of real analytic symplectic diffeomorphisms in the (CC)-case, and, Here it means G -dense in some set of real analytic functions with ﬁxed radius of convergence. This phenomenon is even “prevalent” as shown by Pérez-Marco [36]. If Z converges the observables r ◦ Z , j = 1,... , d are a complete set of real analytic and functionally indepen- ∞ j ∞ dent Poisson commuting integrals. By which we mean the coexistence of quasi-periodic motions and hyperbolic behavior in any neighborhood of the equilibrium; see for a global view on these topics and references the book [2]. We shall in fact see in this paper that there are such dynamical obstructions. Like the accumulation at the origin of hyperbolic periodic points or normally hyperbolic tori. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 7 both in the Hamiltonian or diffeomorphism framework, it can be extended to the (AA)- case (but under the stronger assumption that ω is Diophantine); cf. Theorem 1.3 of our paper. Pérez-Marco’s argument is not based on an analysis of the dynamics of f but rather focuses on the coefﬁcients of the BNF and exploits their polynomial dependence on the coefﬁcients of the initial perturbation by using techniques from potential theory. The following two Theorems are an answer (in the symplectomorphism setting) to Eliasson’s question and decide which of the two assertions of Pérez-Marco’s alternative holds (see Theorem D of Section 1.4 for a more precise statement). Main Theorem 1 ((CC)-Case).— For any d ≥ 1 and any non-resonant frequency vector d d d ω ∈ R , there exists a “prevalent” set of real analytic symplectic diffeomorphism f : (R × R , (0, 0)) ý of the form (1.5) the Birkhoff Normal Forms of which are divergent. In the Action-Angle Case (1.6) it takes the following form: Main Theorem 1’ ((AA)-Case).— For any d ≥ 1 and any Diophantine frequency vector d d ω ∈ R , there exists a “prevalent” set of real analytic symplectic diffeomorphism f : ((R/2π Z) × R , T ) of the form (1.6) the Birkhoff Normal Forms of which are divergent. Main Theorems 1, 1’ also extend to the Hamiltonian case (1.3)–(1.4) (with d = d + 1). Note that from Pérez-Marco’s Theorem (and its analogue in the symplectomor- phism case), in order to prove that the divergence of the Birkhoff Normal Form holds in a prevalent way, it is enough to provide, for each ﬁxed frequency vector ω, one example for which the BNF is divergent. On the other hand, if one is able to construct one such example for some d , then it is easy to construct other such examples for any d > d (see 0 0 for example the proof of Theorem D). Proving Main Theorem 1 (resp. 1’)thusamounts to constructing when d = 1, for each irrational (resp. Diophantine) ω ∈ R, one example of a real analytic symplectic diffeomorphism with a diverging BNF. Gong already provided in [17] (by a direct analysis of the coefﬁcients of the BNF) 2 2 3 examples of real analytic Hamiltonians ω, r+ F : (R × R , 0) → R,F = O (x, y),with Liouvillian frequency ω ∈ R at the origin and a divergent BNF and Yin [54] produced analogue of Gong’s examples in the diffeomorphism case (area preserving map of (R , 0) with a very Liouvillian elliptic ﬁxed point). In these examples the divergence of the BNF is caused by the presence of very small denominators (due to the Liouvillian character of ω) appearing in the coefﬁcients of the BNF. After our result was announced, Fayad [15]con- structed simple examples of real analytic Hamiltonian systems in (R , 0) (d = d + 1 = 4 The idea of using potential theory in problems of small denominators was ﬁrst introduced by Yu. Ilyashenko [23]. See [35] for further references. It is not clear whether one can, for general systems, deduce the case of Hamiltonian ﬂows from the case of diffeomorphisms and vice versa. On the other hand the proofs of Main Theorems 1, 1’ and in particular the proofs of Main Theorem 2 and of Theorems A–B, A’–B’ below extend to Hamiltonian ﬂows with 1 + 1 degrees of freedom. 8 RAPHAËL KRIKORIAN degrees of freedom) with any ﬁxed non-resonant frequency vector at the origin and diver- gent BNF. The argument again is based on an analysis of the coefﬁcients of the BNF; one considers Hamiltonians with two degrees of freedom where two extra action variables are added as formal parameters, one of them appearing later in the denominators of the BNF. These types of examples can be constructed in the diffeomorphism case for d ≥ 3. In a different context, that of reversible systems, let us mention a result of divergence of normal forms in [19] based on a different method (control of coefﬁcients growth) and a result of divergence of normalizing transformations in [33]. We now formulate Eliasson’s question in a stronger form: Question B. — Does the convergence of a formal conjugacy invariant like the Birkhoff Normal Form of a real analytic symplectic diffeomorphism (or Hamiltonian) have consequences on the dynamics of the diffeomorphism (or Hamiltonian)? Note that the convergence of the normalizing transformation has an obvious con- sequence, namely, integrability. As for Question B, there are various results pointing to some kind of rigidity phenomena if analyticity (and some arithmetic properties on ω)is assumed. To be more speciﬁc, let us mention a striking one: Bruno [7] and Rüssmann [42]provedthatif f is real analytic and if its BNF is trivial,BNF(f ) = 2πω, r (in par- ticular BNF(f ) converges), then f is real analytically conjugated to , provided the 2πω,r frequency vector at the origin ω satisﬁes a Diophantine condition. We refer to [53], [25], [11], [9], [51], [18], [13], [12] for generalizations of the Bruno-Rüssmann Theorem and related results. The Main Result of our paper is in some sense one answer, amongst possibly oth- ers, to the previous question at least when d = 1 and if f is assumed to satisfy some twist condition. Let us say that a diffeomorphism of the form (R , 0) ý (1.1)or (R × T, T ) ý is twist (or satisﬁes a twist condition) if the second order term of its BNF is not zero: −1 2 3 (2π) BNF(f )(r) = ωr + b r + O(r ), b = 0. 2 2 Main Theorem 2. — If the Birkhoff Normal Form of a real analytic symplectic twist diffeomor- phism (R , 0) ý (1.1)or (R × T, T ) ý (1.2) converges then the measure of the complement of the union of all invariant curves accumulating the origin is much smaller than what it is for a general such diffeomorphism. In other words, the convergence of a formal object like the BNF has consequences on the dynamics of the diffeomorphism. Precise statements are given in Theorems A–B, A’–B’)ofSection 1.2 and Theorems E and E’ of Section 1.4. Combined with (the exten- sion to the diffeomorphism case of) Pérez-Marco’s Theorem [36], this gives that in any number of degrees of freedom, a general real analytic symplectic diffeomorphism admit- An easily checkable condition. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 9 ting the origin as an non-resonant elliptic equilibrium has a divergent Birkhoff Normal Form (see Theorem D). Having in mind the aforementioned result by Bruno and Rüssmann, a natural stronger question is whether the following rigidity result is true: Question C. — Is it true that a real analytic symplectic diffeomorphism or Hamiltonian system having a Diophantine elliptic equilibrium and a non degenerate and convergent BNF is (real analytically) integrable (in some neighborhood of the origin)? The examples by Farré and Fayad in [14] of real analytic Hamiltonians on d+1 d+1 T × R with convergent BNF and with an unstable Diophantine elliptic torus show that such a generalization is not true for d ≥ 2, at least in the (AA) case and if by non de- generate we mean that BNF(f ) is not trivial. The question is still open for d < 2. Note that though in Farré-Fayad’s examples the BNF (which is explicit) is not degenerate (the rank of its quadratic part is not zero unlike in the Bruno-Rüssmann Theorem), its quadratic part is not of maximal rank. If one drops in Question C the Diophantine assumption and assumes the BNF to be trivial (like in Bruno-Rüssmann’s Theorem) the question is open (this question is related to a question of Birkhoff on pseudo-rotations and to the problem of constructing real analytic Anosov-Katok examples; cf. [16] for details and references). When d = 1 the situation might be more favorable. To any twist area preserving diffeomorphism f : (R , 0) ý (1.1)or f : (R × T, T ) ý (1.2) one can associate (we use the notations and terminology of [46]) its minimal action α : I → R (I is an open interval containing ω) that assigns to each ϕ ∈ I the average action of any minimal orbit with ro- tation number ϕ. The function α is strictly convex (in fact differentiable at any irrational) ∗ 16 and one can thus deﬁne its Legendre conjugate function α : r → sup (ϕr − α(ϕ)) ϕ∈I (see [30], [31], [46] for further details). The function r → α (r) (deﬁned on a neighbor- hood of 0) can be seen as a frequency map in the sense that if γ is an invariant circle for f with “symplectic height” (area with respect to the origin) c then α (c) is the rotation number of f restricted on γ . It has the following properties: the Taylor series of α at 0 coincides with the Birkhoff Normal Form of f ;moreover, if α (hence α) is differentiable 0 0 then f is C -integrable (see [46]). This C -integrability often yields rigidity (we refer to [3], [27] for an illustration of this fact in the context of billiard maps). The techniques de- veloped in our paper are probably enough to prove that if the function α is real analytic then f is in fact real analytically integrable. A more delicate issue is to establish real an- alyticity of α by only knowing that its Taylor series at 0 (the BNF) deﬁnes a converging series. Note that if f is real analytic one can construct dynamically relevant holomor- phic functions (frequency maps) deﬁned on complex domains having positive Lebesgue measure intersections (Cantor sets) with the real axis (see [8], [38]) and which coincide It seems that the (CC) case is not yet settled. Area preserving maps with no periodic points except the origin. 16 ∗ The functions α and α (also denoted β and α) are called Mather’s functions. 10 RAPHAËL KRIKORIAN on these intersections with α . The restrictions of these holomorphic functions on these Cantor sets have some quasi-analyticity properties but it seems that there are not strong enough to deduce that α behaves like a genuine quasi-analytic function (in particular that the convergence of the Taylor series at 0 implies analyticity); we refer to [8]for references and for more details. We conclude this subsection by the following question. Question D. — Is a given real analytic symplectic diffeomorphism accumulated by real analytic symplectic diffeomorphisms having convergent BNF’s? (We do not ask the radii of convergence of the BNF’s to be bounded below). Positive answers to Questions C, D would imply that any real analytic symplectic diffeomorphism admitting an elliptic equilibrium set is accumulated in the strong real analytic topology by diffeomorphisms of the same type that are in addition integrable in a neighborhood of the equilibrium set. 1.2. Invariant circles. — As suggest (1.7), (1.11) the BNF (more precisely its approx- imate version B ) is, as we have already mentioned, a precious tool to study the problem of the existence of quasi-periodic motions in the neighborhood of an elliptic equilibrium. A bright illustration of this fact is certainly the KAM Theorem ([29], [1], [32]) that yields, under suitable non-degeneracy conditions on the BNF (non-planarity), the existence of many KAM tori accumulating the origin (see [13], [12] for results under much weaker non- degeneracy assumptions). We shall be mainly concerned with the 2-dimensional case (d = 1) and we restrict to this case in this subsection. Recall our notation T = R/2π Z for the 1-dimensional torus. An invariant circle (or also invariant curve) for a real analytic (or smooth) diffeomor- phism f : (R × R,(0, 0)) ý of the form (1.5) is the image γ = g(T) of an injective 1 2 C map g : T → R {0} with index ±1at0such that f (γ ) = γ . Likewise, in the (AA) case, an invariant circle or invariant curve) for a real analytic (or smooth) diffeomor- phism f : (T × R, T ) ý of the form (1.6) is the image γ = g(T) of an injective C map g : T → T × R which is homotopic to the circle T = T×{0} and such that f (γ ) = γ . Note that in this latter case, by a theorem of Birkhoff (cf. [4], [21]), invariant circles close enough to T are in fact graphs if f satisﬁes a twist condition: −1 2 (1.14) b (f ) =0if (2π) BNF(f ) = ωr + b (f )r +··· . 2 2 This means that if the given diffeomorphism f has a holomorphic extension to some complex domain W there exists a slightly smaller subdomain W ⊂ W and a sequence of real-symmetric holomorphic diffeomorphisms f deﬁned on Wsuch thatlim sup |f − f |= 0. n→∞ n A KAM torus is an invariant Lagrangian torus on which the dynamics is conjugated to a linear translation with a Diophantine frequency vector. These curves are also called essential curves. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 11 In both cases, we denote by G the set of f -invariant curves and, for t > 0, by L (t) the f f set of points in M := R or T × R which belong to an invariant curve γ ∈ G such that R f γ ⊂ M ∩{|r| < t} L (t) = γ. γ∈G γ⊂M ∩{|r|<t} We then deﬁne (M = R × R or T × R) m (t) = Leb ((M ∩{|r| < t}) L (2t)). f M R f Notation 1.1. — We shall use the following notations: if a ≥ 0 and b > 0 are two real numbers we write a b for: “there exists a constant C > 0 independent of a and b such that a ≤ Cb”. If we want to insist on the fact that this constant C depends on a quantity β we write a b. We shall also write a b to say that a/b is small enough and a b to express the fact that this smallness condition depends on β . The notations b a, b a, b aand b a are deﬁned in the same way. When β β one has a band b a we write a b. The 2-dimensional version of the KAM Theorem is the celebrated Moser’s twist Theorem [32] (see also [43]): Theorem (Moser).— Let f be a symplectic smooth diffeomorphism like (1.5)or(1.6) satisfying the twist condition (1.14). If f admits Birkhoff Normal Forms at the origin to all orders then, for any constant a > 0, (1.15) m (t) t . Let us comment on the previous result. In the (CC) case, it is in fact enough to as- sume that ω in (1.5) is non-resonant since under this condition f admits Birkhoff Normal Forms to all orders. On the other hand, in the (AA) case the existence of the BNF to all orders, more precisely the existence of solutions to the related cohomological equations (see Lemma E.7), requires ω in (1.6) to be Diophantine; in the real analytic case a weaker arithmetic condition is enough, see [44], [45]. If in the (CC) case ω is non-resonant only up to some low order, (1.15) holds only for some a > 0 (see [38]). If in the (AA) case we drop the assumption that ω is Diophantine (but we assume ω to be non-resonant) then, though no BNF is available, m (t) in (1.15)goestozeroas t goes to zero but not necessarily as a power of t: indeed, in the corresponding (AA) case of sufﬁciently smooth Hamiltonian systems, Bounemoura proves in [6], in any number of degrees of freedom In the (AA) case M ∩{|r| < t}={(θ , r) ∈ T × R, |r| < t} and in the (CC) case M ∩{|r| < t}={(x, y) ∈ R R 2 2 R × R,(1/2)(x + y )< t}. This means one can deﬁne B (f ) for any N ≥ 2. If we denote p /q the convergents of ω,itreads ln q = o(q ). In comparison, the classical Diophantine condition n n n+1 n amounts to ln q = O(ln q ). n+1 n 12 RAPHAËL KRIKORIAN and under a Kolmogorov non-degeneracy condition, that the origin is KAM stable (see [13] for a previous similar result in two degrees of freedom and in the real analytic case) and provides measure estimates for the complement of the set of the invariant tori. Let us add that the twist condition (1.14) in Moser’s Theorem can be considerably weak- ened (see for example [13], [12]). When d = 1, symplecticity (area preservation) can be replaced by the weaker intersection property. When f is real analytic and ω (both in the (CC) and (AA) cases) is Diophantine one can get, by pushing to its limit the “standard” KAM method, a better estimate: for any 0 <β 1and t 1 one has −β 1+τ(ω) (1.16) m (t) exp(−(1/t) ) where we have deﬁned for any irrational ω − ln min |kω − l| ln q l∈Z n+1 (1.17) τ(ω) = lim sup = lim sup ≥ 1; ln k ln q k→∞ n→∞ n in the preceding formula (p /q ) is the sequence of convergents of ω.Notethatif n n n≥0 τ(ω) < ∞,then ω is Diophantine with exponent τ for any τ >τ(ω) (cf. (1.10)). In this case, the inequality (1.16) is known to be true with the exponent 1/(1 + τ ) on the right hand side (see for example [26] and the references therein). If τ(ω)=∞ we say that ω is Liouvillian. 1.3. Optimal and improved measure estimates. — The main results of our paper are that: (A) one can improve the exponent in (1.16)if BNF(f ) converges; (B) in the “general case” the exponent in (1.16) is almost optimal. More precisely Theorem A. — Let f be a real analytic symplectic diffeomorphism f : (R × R,(0, 0)) ý like (1.5)or f : (T × R, T ) ý like (1.6) satisfying the twist condition (1.14) and assume that in both cases ω is Diophantine. Then, if BNF(f ) deﬁnes a converging series one has for any 0 <β 1 and 0 < t 1 (1/τ (ω))−β (1.18) m (t) exp − . On the other hand general real analytic twist symplectic diffeomorphisms like (1.5), (1.6) behave quite differently: Theorem B. — Let ω ∈ R be Diophantine. There exist real analytic twist symplectic diffeo- morphisms f : (R × R,(0, 0)) ý like (1.5)or f : (T × R, T ) ý like (1.6) satisfying the twist I.e. accumulated by a positive measure set of invariant quasi-periodic tori. 24 ∗ As usual, if ω = 1/(a + 1/(a + 1/(··· ))), a ∈ N ,wedeﬁne p /q = 1/(a + 1/(a + 1/(···+ 1/a )). 1 2 i n n 1 2 n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 13 condition (1.14) and a sequence of positive numbers (t ) converging to zero such that for any 0 <β 1, 0 < t 1 k β ( )+β 1+τ(ω) (1.19) m (t ) exp − . f k As we already mentioned in the previous subsection, when ω is very Liouvillian, −1 for example when lim inf (q ln q )> 0, it is not clear, in the (AA) case, how to deﬁne n n+1 BNF(f ). On the other hand, in the (CC) case, BNF(f ) is deﬁned whenever ω is non- resonant and as we will soon see in Theorem A’ below, the result of Theorem A extends to this situation. One might still wonder whether a weaker Diophantine condition ln q = n+1 o(q ) (or something slightly stronger) is enough to ensure the validity of Theorem A in the (AA) case (remember that in this case, BNF(f ) is well deﬁned). It seems possible that adapting Propositions 5.3, 5.5 to this situation (using e.g. [44], [45]) provides estimates that are still good enough to make the proof of Theorem A work. Let us deﬁne for ω ∈ R Q −1 5min(|b (f )|,|b (f )| ) 2 2 (1.20) t (ω) = . q q n+1 n Theorem A’. — Let ω be Liouvillian and f : (R× R,(0, 0)) ý be a real analytic symplectic diffeomorphism of the form (1.5) satisfying the twist condition (1.14). Then, if BNF(f ) deﬁnes a converging series, one has for every k ∈ N large enough such that q ≥ q k+1 1/5 (1.21) m (t (ω)) exp(−q ). f k k+1 Note that if ω is Liouvillian, one has for inﬁnitely many k, q ≥ q . k+1 On the other hand: Theorem B’. — For any ω ∈ R Q, there exist real analytic symplectic diffeomorphisms f : (R × R,(0, 0)) ý of the form (1.5) satisfying the twist condition (1.14) such that for every β> 0 and inﬁnitely many k ∈ N (1.22) m (t (ω)) exp(−q ). f k k+1 Theorem A is a consequence of the following theorem: Theorem C (Small holes).— Let ω be Diophantine and let f : (R× R,(0, 0)) ý of the form (1.5)or f : (T × R, T ) ý of the form (1.6) be a real analytic symplectic diffeomorphisms satisfying the twist condition (1.14). Then, for t > 0, there exists a ﬁnite collection D of pairwise disjoint disks D of the complex plane centered on the real axis such that, for any 0 <β 1, 0 < t 1 one has: q q (1) The number #D of disks in the collection D satisﬁes t t 1−β (1.23)#D (1/t) t 14 RAPHAËL KRIKORIAN and one has −β q q q 1+τ(ω) (1.24) ∀ D ∈ D |D ∩ R| exp(−(1/t) ) (1/τ (ω))−β (1.25) m (t) exp(−(1/t) ) + |D ∩ R|. q q D∈D q q (2) If BNF(f ) converges, then for any t 1 one has for each D ∈ D β t (1/τ (ω))−β (1.26) |D ∩ R| exp(−(1/t) ). Estimate (1.24) explains why in general (without the assumption that BNF(f ) con- verges) one only gets the estimate (1.16). We shall explain in Section 1.5.1 where these disks Dcome from. There is a corresponding theorem in the Liouvillian (CC) case that implies The- orem A’. We shall not state it but we mention that it is a consequence of Theorem 12.6 and Corollary 13.6. Theorems A, A’, C are proved in Section 14 as consequences of Theorems 12.3, 12.6 and Corollaries 13.2, 13.6. Theorems B and B’ are consequences of Theorems E and E’ which are stated in the next Section 1.4. These Theorems are proved in Section 16 which uses results from Section 15. Because 1/τ > 1/(1 + τ), Theorems A–B, A’–B’ clearly imply, in the (AA) and (CC) case, when d = 1, the existence of a diffeomorphism f of the form (1.5)–(1.6) with divergent BNF. We explain in the next Section 1.4, see Theorem D, that this implies Main Theorems 1–1’ in the elliptic ﬁxed point case and the action-angle case for any d ≥ 1 and in a prevalent way. 1.4. Prevalence of divergent BNF’s. 1.4.1. The Dichotomy Theorem. — Let us explain more precisely the dichotomy of R. Pérez-Marco mentioned in Section 1.1 Deﬁnition 1.2. — Asubset A of a real afﬁne space E is (PM)-prevalent if there exists F ∈ A such that for any F ∈ E the set {t ∈ R, tF + (1 − t)F ∈ / A} has 0 Lebesgue measure. 0 0 Pérez-Marco’s dichotomy for Hamiltonians having a non-resonant elliptic ﬁxed point can be reformulated the following way: let E be theafﬁne spaceofrealanalytic See [22] for the concept of prevalence. We can replace zero Lebesgue measure by zero (logarithmic) capacity like in Pérez-Marco’s paper. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 15 Hamiltonians 2 2 3 H(x, y) = 2π ω (x + y )/2 + F(x, y), F(x, y) = O (x, y) j j j=1 which are perturbations of a given non-resonant quadratic part 2 2 (r) = 2πω, r= 2π ω (x + y )/2 ω j j j j=1 and let A be the set of those Hamiltonians which have a divergent BNF. Then, Pérez- Marco’s dichotomy is: either for any H ∈ E ,BNF(H) converges or A is (PM)-prevalent. ω ω We now discuss the extension of Pérez-Marco’s dichotomy to the case of symplectic diffeomorphisms in the (AA) and (CC)-cases. d d Any real analytic symplectic diffeomorphism f : (R × R ,(0, 0)) ý of the form d d (1.5)or f : (T × R , T ) ý of the form (1.6) can be parametrized in the following con- venient form: (1.27) f = ◦ f , 2πω,r F d d 3 d d 2 where, F : (R × R ,(0, 0)) → R,F = O (x, y) or F : (T × R , T ) → R,F = O (r) is some real analytic function and where we denote f : (x, y) → ( x, y) or (θ , r) → (θ, r) the exact-symplectic map (see Section 4.5) deﬁned implicitly by x = x + ∂ F(x, y), y = y + ∂ F(x, y)(CC case) y x (1.28) or θ = θ + ∂ F(θ , r), r = r + ∂ F(θ , r)(AA case). r θ d d d d d For d ≥ 1, ω ∈ R non-resonant, we deﬁne S (R × R ) (resp. S (T × R )) the set of real ω ω d d d d analytic symplectic diffeomorphisms f : (R × R ,(0, 0)) ý (resp. f : (T × R , T ) ý) d d 3 d of the form f = ◦ f with F : (R × R ,(0, 0)) → R,F = O (x, y) (resp. F : (T × 2πω,r F d 2 d d R , T ) → R,F = O (r)) real analytic. We then say that a subset of S (R × R ) (resp. 0 ω d d S (T × R ))is (PM)-prevalent if it is of the form { ◦ f , F ∈ A} for some (PM)- ω 2πω,r F ω d d 3 ω d d 2 prevalent subset A of C (R × R , R) ∩ O (x, y) (resp. C (T × R , R) ∩ O (r)). Here is the version of Pérez-Marco’s Dichotomy Theorem [36] for real analytic symplectic diffeomorphisms of the 2d -disk or the 2d -cylinder. Theorem 1.3 (Dichotomy Theorem).— Let d ≥ 1 and ω ∈ R be a non-resonant frequency d d vector. Then, either for any f ∈ S (R × R ), the formal series BNF(f ) converges (i.e. the series it d d deﬁnes has a positive radius of convergence), or there exists a (PM)-prevalent subset of S (R × R ) such that for any f in this subset BNF(f ) diverges. d d The same dichotomy holds in S (T × R ) provided ω is Diophantine. ω 16 RAPHAËL KRIKORIAN As we mentioned earlier Pérez-Marco’s Dichotomy Theorem was proved in the setting of real analytic Hamiltonians having an elliptic ﬁxed point. Its extension to the dif- feomorphism setting follows essentially Pérez-Marco’s arguments. We refer to Section 6.2 for further details in particular in the Action-Angle case (cf. Lemma 6.3). 1.4.2. Prevalence of the divergence of the BNF: Main Theorems 1, 1’.— As a Corollary of Theorem 1.3 we now obtain, using Theorems A and B,Theorems A’ and B’,the following precise formulation of Main Theorems 1, 1’: d d d Theorem D. — For any d ≥ 1 and any non-resonant ω ∈ R ,the setof f ∈ S (R × R ) d d with a divergent BNF is (PM)-prevalent. If ω is Diophantine the same result holds with S (T × R ) d d in place of S (R × R ). Proof. — We give the proof in the case of real analytic symplectic diffeomorphisms of the 2d -disk. Let ω = (ω ,...,ω ) ∈ R be non-resonant. According to Pérez-Marco’s di- 1 d chotomy (Theorem 1.3) it is enough to provide one example of a real analytic symplectic diffeomorphism of the 2d -disk with diverging BNF and frequency vector ω at the ori- gin to get the conclusion. Since ω is non-resonant, there exists 1 ≤ j ≤ d such that ω is irrational. According to whether ω is Diophantine or Liouvillian we use Theorems A and B or Theorems A’ and B’ to produce a real analytic symplectic diffeomorphism f : (R , 0) ý with frequency ω at the origin and with a divergent BNF. We now deﬁne j j d d f : (R × R ,(0, 0)) ý by f (x ,..., x , y ,..., y ) = ( x ,... x , y ,..., y ), 1 d 1 d 1 d 1 d √ √ 2π −1ω for k = j,( x + −1 y ) = e (x + −1y ) k k k k ( x , y ) = f (x , y ). j j j j j This diffeomorphism is real analytic, symplectic and BNF(f )(r ,..., r ) = BNF(f )(r ) + 2πω r 1 d j j k k k∈{1,...,d}j is diverging since BNF(f ) is. 1.4.3. Prevalence of optimal estimates: Main Theorem 2.— We now present two theo- rems (Theorems E and E’) stating that the measure estimates (1.19)ofTheorem B and (1.22)ofTheorem B’ are prevalent. Together with Theorems A, A’ and the fact that 1/(τ + 1)< 1/τ , this gives a more precise meaning to our Main Theorem 2. We shall treat the (AA) and (CC) cases separately. 2 N 2 ∗ Let X be the set ([−1, 1] ) ={(ζ ,ζ )∈[−1, 1] , k ∈ N } endowed with the 1,k 2,k ⊗N product measure μ = (Leb 2 ) . ∞ [−1,1] ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 17 (AA) Case. Let f = ◦ f 2 be a real analytic symplectic twist map of the annulus of 2πωr O(r ) the form (1.6) and satisfying the twist condition (1.14). For ζ ∈ X and h > 0we deﬁne G ∈ C (T × R) (h > 0ﬁxed) a −|k|h G (θ , r) = r e (ζ cos(kθ) + ζ sin(kθ)) ζ 1,k 2,k k∈N where a is some universal integer (appearing in Proposition G.1 of Appendix G). Theorem E ((AA) case).— For any Diophantine ω and for any 0 <β 1, there exists an inﬁnite set N ⊂ N such that if t = t (ω) is the sequence deﬁned by (1.20), then for μ -almost β k k ∞ ζ ∈ X , the estimate (1.19)ofTheorem B with f replaced by f is satisﬁed for inﬁnitely many k ∈ N . ζ β In particular, using Theorem A,the BNFof f := f ◦ f is divergent for μ -almost ζ ∈ X . ζ G ∞ (CC) Case. We formulate the corresponding result in the (CC) Case in a less general setting than in the (AA) Case. We assume that 2 2 a f = 2 2 + O((x + y ) ) ((1/2)(x +y )) where −1 2 3 (2π) (r) = ωr + b r + O(r ), b = 0. 2 2 We shall denote sign(b )=±1if ±b > 0. 2 2 For ζ ∈ X ,let G be the real analytic function a q q 2 2 3 k k x + y ζ x + iy x − iy 1,k ∗ −iπ/4 iπ/4 G (x, y) = × √ e + √ e 2 2 2 2 k=1 q q k k ζ x + iy x − iy 2,k −iπ/4 iπ/4 + × √ e − √ e 2i 2 2 and f = f ◦ . ζ G Theorem E’ ((CC) Case).— For any non-resonant (resp. Diophantine) ω and any 0 <β 1, there exist a non-empty set s(ω) ∈{−1, 1} (resp. s (ω) ∈{−1, 1}) and an inﬁnite set N ⊂ N (resp. N ⊂ N) such that the following holds. If sign(b ) ∈ s(ω) (resp. sign(b ) ∈ s (ω)), then, for 2 2 β μ -almost ζ ∈ X , the estimate (1.22)ofTheorem B’ (resp. (1.19)ofTheorem B) with f replaced by f is satisﬁed for inﬁnitely many k ∈ N (resp. k ∈ N ). In particular, using Theorems A, A’, for any non-resonant ω,the BNFof f := f ◦ is divergent for μ -almost ζ ∈ X . ζ G ∞ We refer to Section 16 for the proof of Theorems E and E’. In the non-resonant case, the sets s(ω) and N ⊂ N do not depend on β . 18 RAPHAËL KRIKORIAN 1.5. Some words on the proofs. — The starting point of the proofs of Theorems A, A’,and C is a KAM scheme that we implement on a holomorphic extension of the real analytic diffeomorphism f . This allows to work with holomorphic functions deﬁned on complex domains “with holes” (i.e. disks which are removed). If these domains are “nice” we can use some quantitative form of the analytic continuation principle to propagate informations in the neighborhood of the origin, like the convergence of the BNF, to the neighborhoods of each hole. We illustrate this with the proof of Theorem C. 1.5.1. Sketch of the proof of Theorem C.— We describe it in the (AA) case. Let f : (T × R, T ) ý, T = T×{0}, be a real analytic symplectic diffeomorphism of the form 0 0 (1.27)with b (f ) = 0and ω Diophantine. After performing some steps of the Birkhoff Normal Form procedure mentioned in the introduction, we can assume that −1 2 m (1.29) f = ◦ f ,(2π) (r) = ωr + b r +··· , F = O(r ) F 2 where m is large enough and where f is the exact symplectic map (cf. (1.28)) associated to some real symmetric holomorphic function F : T × D(0, ρ) → C (h,ρ> 0); the notations T , D(0, ρ) are for T := ((R+ i]− h, h[)/(2π Z)), D(0, ρ)={r ∈ C, |r| < ρ}. h h Adapted KAM Normal Form. — Theorem C can be seen as an improved version of the classic KAM Theorem on the positive Lebesgue measure of the set of points lying on invariant curves (cf. Moser’s Theorem of Section 1.2). There are several ways to prove this standard KAM Theorem. A direct approach (which goes back to Arnold in his proof of Kolmogorov’s theorem) is to ﬁnd a sequence of (real symmetric) holomorphic symplectic diffeomorphisms g close to the identity, deﬁned on smaller and smaller complex domains −1 T × U (h ≥ h ≥ h/2, U ⊂ U ⊂ D(0, ρ))and such that g ◦ f ◦ g gets closer and h i i−1 i i i−1 i i i closer to some integrable models : −1 (1.30) [T × U ] g ◦ f ◦ g = ◦ f , F 1 h i i F i i i i i (in the preceding formula, the set written on the left is a domain where the conjugation relation holds); see Figure 1. One then proves that g and converge (in some sense) i i on T × (U ∩ R) (U := U ) to some limits g , and that U ∩ R (in general a ∞ ∞ i ∞ ∞ ∞ Cantor set) has positive Lebesgue measure. The searched for set of f -invariant curves is then g ({r = c}) and one has for some ﬁxed constant a > 0and any ρ< ρ c∈U ∞∩R (1.31) m (ρ) F . We refer to Theorem 12.1 for more details. The domains U can be chosen to be holed domains i.e. disks D(0,ρ )(ρ ≈ ρ ) from which a ﬁnite number of small complex disks i i This means that it takes real values when θ and r are real. This means that depends only on the r variable. i ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 19 D(0, ρ) R-axis KAM FIG. 1. — The holed domains U where the KAM-Normal Form U is deﬁned (the holes, caused by resonances, are the grey disks) centered on the real axis (the “holes” of U ) have been removed. Removing these small disks is due to the necessity of avoiding resonances when one inductively construct g , , F i i i from g , , F . More precisely, U is essentially obtained from U by removing i−1 i−1 i−1 i i−1 −1 “resonant disks” i.e. disks where the “frequency map” (2π) ∂ is close to a ratio- i−1 nal number of the form l/k, (l, k) ∈ Z × N ,max(|l|, k) N (N is an exponentially i−1 i increasing (in i) sequence which is deﬁned at the beginning of the inductive procedure). The sizes of the holes of U created by removing a ﬁnite number of disks from U decay i i−1 very fast with i. We shall call a conjugation relation like (1.30) a(n) (approximate) KAM Normal Form for f . Its construction is presented in Section 7. A useful observation (cf. Section 10) is that, depending on ρ< ρ , one can choose indices i (ρ) < i (ρ) such that all the holes D of the domain U that intersect D(0,ρ), − + i (ρ) are disjoint and are created at some step i − 1 = i ∈[i (ρ), i (ρ)] (hence D ⊂ U ); D − + i moreover, i (ρ) is large enough to ensure that the size of D is small. Writing (1.30)with i = i (ρ) we get (note the change of notations) KAM −1 KAM KAM KAM (1.32) [T × U ] g ◦ f ◦ g = ◦ f , F 1. h/2 i (ρ) i (ρ) i (ρ) + i (ρ) + + + i (ρ) i (ρ) + + This is what we call our adapted KAM Normal Form (adapted to D(0,ρ)); see Section 10. With the choice we make for i (ρ) we have KAM (1/τ )− (1.33) F exp(−(1/ρ) ), i (ρ) KAM (1/τ )−β where the last formula means: “for any β> 0, F exp(−(1/ρ) )”. i (ρ) Hamilton-Jacobi Normal Forms. — Cf. Section 8.Ahole D ⊂ U of the domain U that i i (ρ) D + −1 KAM is created at step i corresponds as we have mentioned to a resonance (2π) ∂ ≈ l/k, (l, k) ∈ Z × N ,max(|l|, k) N that appears when one constructs the KAM Nor- mal Form (1.30)fromstep i to step i + 1. In this resonant situation we are able to D D 20 RAPHAËL KRIKORIAN HJ −1 KAM FIG. 2. — Hamilton-Jacobi Normal Form close to a resonance (2π) ∂ (c) = l/k. The holomorphic function D i HJ is deﬁned on the annulus D D associate to D a Hamilton-Jacobi Normal Form, cf. Section 8,Proposition 8.1: there exists an q q q annulus D D(D, D are disks), D ⊂ U , D ⊃ D, D ⊃ D(D is small but much bigger than D) on which one has HJ HJ −1 HJ HJ (1.34) [T × D D] (g ) ◦ ◦ f ◦ (g ) = ◦ f , h/9 F D D i i F D D D D HJ KAM (1.35) F F . D i (ρ) See Figure 2. This HJ Normal Form also satisﬁes the important Extension Property which in q q some situation allows to bound above the size of D (note that in general the sizes of Dand HJ D are comparable). It states that if the holomorphic function , which is deﬁned on the annulus D D, coincides to some very good order of approximation with a bounded holomorphic function deﬁned on the disk D, then D can be chosen to be small (see the quantitative statement of Proposition 8.1). Proof of the ﬁrst part (1.25)ofTheorem C.— Applying the aforementioned standard KAM estimate (1.31) on the holed domain U to ◦ f KAM (cf. (1.32)) and on each annulus i (ρ) + i (ρ) F + i (ρ) D Dto HJ ◦ f HJ,(cf. (1.34)) together with the estimate (1.35) we get that outside a D D set of measure |D ∩ R| the invariant curves of f cover a set the complement of D∈D KAM a which in D(0,ρ) has a measure F for some a > 0; hence the inequality (1.25) i (ρ) by (1.33). For more details see the proof of Theorem 12.3. Birkhoff Normal Forms. — Cf. Section 6. To prove the second part of Theorem C,(1.26)we need to introduce one further approximate Normal Form, namely the approximate Birkhoff b b KAM τ τ Normal Form (cf. Section 6)valid on T × D(0,ρ ) (b = τ + 1), D(0,ρ ) ⊂ U h/2 τ i (ρ) b BNF −1 BNF (1.36) [T × D(0,ρ )] (g ) ◦ f ◦ (g ) = BNF ◦ f BNF , h/2 F ρ ρ ρ ρ BNF KAM (1.37) F F . ρ i (ρ) See Figure 3. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 21 KAM D(0,ρ) i (ρ) D(0,ρ ) R-axis BNF b FIG. 3. — The approximate Birkhoff Normal Form is deﬁned on D(0,ρ ). It coexists with the KAM Normal Form KAM deﬁned on the holed domain U i (ρ) i (ρ) + KAM D(0,ρ) i (ρ) D(0,ρ ) R-axis D, D BNF b FIG. 4. — The three Normal Forms. The holomorphic function is deﬁned on D(0,ρ ); associated to each hole D HJ the holomorphic function is deﬁned on the annulus D D. These Normal Form coincide with the KAM Normal KAM KAM Form deﬁned on the holed domain U i (ρ) i (ρ) + + Proof of the second part (1.26)ofTheorem C.— Having the three Normal Forms (1.32), (1.36), (1.34) in hand (see Figure 4) the proof of the second part of Theorem C relies on the following three principles. 22 RAPHAËL KRIKORIAN HJ KAM BNF – Comparison Principle cf. Section 9: since F ,F F are equally very small, all the i (ρ) D ρ previous Normal Forms almost coincide on the intersections of their respective domains of deﬁnition (this is done in Proposition 9.1); more precisely their frequency maps almost coincide HJ BNF KAM (1.38) ρ i (ρ) b KAM KAM τ q D(0,ρ )∩U U ∩(DD) i (ρ) + i (ρ) where the symbol a b (a, b are functions and V is an open set) means here: for all z ∈ V, (1/τ )− |a(z) − b(z)| exp(−(1/ρ) ), ρ and τ being ﬁxed. Moreover, if the formal BNF converges and equals a holomorphic function deﬁned on, say, D(0, 1), one has also (cf. Corollary 6.7) BNF D(0,1)∩D(0,ρ ) b KAM and in particular from (1.38)(we have D(0,ρ ) ⊂ U ) i (ρ) KAM (1.39) . i (ρ) D(0,1)∩D(0,ρ ) KAM b – No-Screening Principle, cf. Section 3: Since and almost coincide on D(0,ρ ) and i (ρ) KAM are holomorphic on the bigger domain D(0, 1)∩ U , one can be tempted to infer that i (ρ) they also almost coincide on this latter domain. A difﬁculty could appear here: an exces- KAM sive number of holes of D(0, 1) ∩ U (in comparison to their sizes) could cause some i (ρ) “screening effect” (like in Electrostatics) that prevents the propagation of the information KAM given by (1.39) to “most of ” the domain D(0, 1)∩ U ; see Section 3.2 for more details. i (ρ) This is the reason why, instead of working on the whole domain D(0, 1) we work on the smaller one D(0,ρ). In this situation, the choice we make for i (ρ) (cf. (10.301)) is such KAM that the number of holes of D(0, 1) ∩ U is not too big in comparison to their sizes; i (ρ) this is studied in Section 10,Proposition 10.4. This allows us to apply Proposition 3.1 and to extend the domain of validity of the approximate equality (1.39) to (a good part KAM of) D(0, 1) ∩ U : i (ρ) KAM (1.40) . i (ρ) KAM D(0,1)∩U i (ρ) – Residue or Extension Principle cf. Section 8.8: From (1.38), (1.40) one has HJ (1.41) KAM U ∩(DD) i (ρ) HJ or,inother words, , which is deﬁned on the annulus D D, coincides with a very good approximation with a holomorphic function deﬁned on the whole disk D. The aforementioned Extension Principle of Proposition 8.1, which essentially amounts to the computation of ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 23 a residue (done in Paragraph 8.8.1), then tells us that the radius of D is much smaller (1/τ )− than what we expected it to be: ﬁnally, |D ∩ R| exp(−(1/ρ) ). This is (1.26). Formoredetails we refertoProposition 10.7 and Corollary 13.2. 1.5.2. On the proof of Theorem C in the elliptic ﬁxed point case. — The proof in the non- resonant elliptic ﬁxed point case, f : (R , 0) ý, follows the same strategy especially if the frequency ω is Diophantine. A technical point is that to be able to implement the KAM BNF No-Screening Principle of Section 3 we need to work with domains U ⊃ U where i (ρ) ρ BNF U is a disk around 0 (the estimate on the analytic capacity of this disk is then favorable). This is the reason why we cannot in this situation use Action-Angle variables since this would force us to work on angular sector domains and not disks. Instead, we deﬁne our approximate BNF and KAM Normal Forms directly in Cartesian Coordinates. The formalism turns out to be the same as in the Action-Angle case (see Section 5), so we treat these two cases simultaneously. The case where ω is Liouvillian is done in a similar (and even simpler) way. 1.5.3. On the proofs of Theorems B, B’, E and E’.— The proofs are more classical and based on the fact that, in the general case, resonances are associated to the existence of hyperbolic periodic points in the neighborhood of which no (“horizontal”) invariant circle can exist. To see this in a special situation (we describe it in the (AA)-case) let f = ◦ f , −1 2 ∗ where R r → (2π) (r) = (p/q)r + br /2 + ··· ∈ R with p ∈ Z, q ∈ N mutually prime and, say, b > 0 (for example b = 1); we also assume that T× R (θ , r) → F(θ , r) = 3 −1 O(r ) ∈ R and is (2π/q)-periodic in θ . The origin is thus resonant since (2π) ∂(0) = p/q is rational. One can approximate ◦ f by per 1 f := (θ , r) → 2π(p/q, 0) + (θ , r) 1 2 where is the time-1 map of the Hamiltonian H(θ , r) = br /2 + F(θ , r). Observe that because we have assumed that F(θ , r) is 2π/q-periodic in θ , the same is true for , hence the maps and (θ , r) → (θ , r) + 2π(p/q, 0) commute; thus, understanding the dynamics of f essentially amounts to understanding that of . This latter dynamics is easy to analyze since it is the time-1 map of a Hamiltonian vector ﬁeld in dimension 2, namely a pendulum on the cylinder T× R. Indeed, if F is “typical”, a change of coordinates leadsustothe case where ∂ F(0, 0) = 0and ∂ F(0, 0)< 0hence H(θ , r) = cst+ br /2− (1/2)|∂ F(0, 0)|θ + h.o.t. Under this form it is clear that (0, 0) is a hyperbolic ﬁxed point for and since H is 2π/q-periodic, the same is true for the points (2π k/q, 0), k = 0,..., q − 1. Since these points are permuted by (θ , r) → (θ , r) + 2π(p/q, 0), this shows that (0, 0) is a hyperbolic q-periodic point for f (this means a hyperbolic ﬁxed q 2 per point for f ). If |∂ F(0, 0)| is not too small compared to the approximation f − f ,the To say it shortly, in Poisson-Jensen’s formula on subharmonic functions (see Section 3.1), the “weight” of a small disk D(0,ρ) ⊂ D(0, 1) is 1/| ln ρ| while the “weight” of D(0,ρ)∩ ⊂ , being an angular sector at 0 is only ρ , a > 0. 24 RAPHAËL KRIKORIAN point (0,0) will also be a q-periodic hyperbolic point for f . However, a horizontal invariant circle cannot cross the stable or invariant manifolds of this periodic point; this establishes the existence of a zone in which horizontal invariant circles cannot pass. To quantify the size of this zone one just has to estimate the strength of the hyperbolicity of the periodic point and the size of the corresponding local stable and unstable manifolds. The more general case where we do not assume a priori that F(θ , r) is 2π/q-periodic nor F(θ , r) = O(r ) can essentially be reduced to the preceding example, provided F is small with respect to 1/q. This requires the use of a resonant normal form described in Appendix G. For “generic” symplectic diffeomorphisms of the form (1.6) satisfying a twist condition (1.14) one can establish the existence of hyperbolic zones associated to any best rational approximation p /q of ω; these zones accumulate the origin. We refer n n to Sections 15 and 16 for more details. 1.6. Organization of the paper. — Section 2 is essentially dedicated to ﬁxing some notations and introducing the notion of domains with holes that plays a central role in the KAM approach (à la Arnold). We discuss Cauchy’s estimates and Whitney’s extension Theorem in this framework. The not so standard notations used in the text are summa- rized in Section 2.6. In Section 3 we give a brief account of what is the screening effect and we provide a no-screening criterion which will be useful for our purpose. It is based on Poisson-Jensen’s formula on subharmonic functions applied in a domain with not too many holes (w.r.t. their sizes). In Section 4 our main purpose is to check that estimates on compositions of gener- ating functions hold in the case of domains with holes. We treat in a uniﬁed way the CC and AA cases. We also discuss invariant curves. In Section 5 we study the (co)homological equations and state a proposition on the basic KAM step (Proposition 5.5). Birkhoff Normal Forms (approximate and formal) are presented in Section 6 and Appendix E. We explain in Section 6.2 how Pérez-Marco’s dichotomy extends to the diffeomorphism case. Section 7 is dedicated to the KAM scheme which is central in our paper; we pay particular attention to the location of the holes of the KAM-domains. In Section 8 we present the Hamilton-Jacobi Normal Form associated to each resonance appearing during the KAM scheme. Their construction is based on a Res- onant Normal Form and an argument of approximation by vector ﬁelds the proofs of which are left in the Appendix, Sections G and H. The most important property of these Hamilton-Jacobi Normal Forms is the Extension Property that states that if the correspond- ing frequency map deﬁned on a annulus is very close to a holomorphic function deﬁned on a bigger disk containing the annulus, the domain of validity of this Normal Form is essentially this disk. The Matching or Comparison Principle is presented in Section 9. It quantiﬁes the fact that (exact) symplectic maps have essentially one frequency map. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 25 Residue/Extension No screening Principle (Section 3) (Section 8) HJ-NF≈ HJ-NF on Dsmall on D D D D (Section 8) (Sections 8, 13) (Section 8) Resonant NF (Appendix G) Resonances, Holes D ∈ D (Section 8) Positive KAM measure procedure estimates (Section 7) (Section 12) Matching KAM NF on Principle U = D(0,ρ) D D∈D (Section 9) (Sections 7, 10)) Formal BNF Improved Approx. BNF converges measure bτ on D(0,ρ ) ≈ = on D(0, 1). estimates (Section 6) (Assumption) (Conclusion) FIG. 5. — Plan of the proof of the improved measure estimate (1.26)ofTheorem C We construct in Section 10 and 11 our coexisting adapted KAM, BNF and HJ Normal Forms in the respective cases ω Diophantine or Liouvillian, the latter being easier to treat. In Section 12 we ﬁrst state a generalization of the classical KAM estimate on the measure of the set of invariant curves that hold on domains with holes (Theorem 12.1) and we apply it to our adapted KAM and HJ Normal forms to get measure estimates on the set of invariant curves lying in the union of the domains of deﬁnitions of these Normal Forms. This provides Theorems 12.3 and 12.6 which play an important role in the proofs of Theorems C, A and A’. 26 RAPHAËL KRIKORIAN In Section 13 we use the Extension Principle of Section 8 to show that if the BNF converges the measure estimates provided by Theorems 12.3 and 12.6 improve consid- erably. In Section 14 we conclude the proofs of Theorems C, A and A’. The mechanism for the creation of zones of the phase space that do not intersect the set of invariant circles is presented in Section 15 (Proposition 15.1). This allows us to construct (prevalent) examples that satisfy Theorems B, B’, E and E’ in Section 16. Finally, an Appendix completes the text by giving more details on the proofs of some statements or by presenting more or less classical methods that had to be adapted to our more speciﬁc situation. 2. Notations, preliminaries Let T be the 1-dimensional torus T := R/(2π Z)={x + 2π Z, x ∈ R} and for 0 ≤ h≤∞ T = T∪{x + iy + (2π Z), x, y ∈ R, |y| < h} (i =−1) the complex cylinder of width 2h.If θ = (x + iy )+ (2π Z), θ = x + iy + (2π Z) ∈ T 1 1 1 2 2 2 ∞ we set |θ − θ | := min |(x − x − 2π l) + i(y − y )|. 1 2 T l∈Z 1 2 1 2 If ρ> 0 wedenoteby D(z,ρ) ⊂ C the open disk of center z and radius ρ and by D(z,ρ) its closure; sometimes for short we shall write D for D(0,ρ) (and by D its ρ ρ closure). If z = x + iy ∈ C,(i = −1) x, y ∈ R, (resp. θ = x + iy + (2π Z) ∈ T ), we denote by z = x − iy (resp. θ = x − iy + (2π Z)) its complex conjugate. 2 2 We deﬁne the involutions σ ,σ : C → C and σ : T × C → T × C by 1 2 3 ∞ ∞ (2.42) σ (x, y) = (x, y), σ (z,w) = (iw, iz), σ (θ , r) = (θ, r). 1 2 3 For w = (w ,w ), w = (w ,w ) ∈ C × C (resp. ∈ T × C) we deﬁne the distance 1 2 ∞ 1 2 d(w, w ) = max(|w − w |,|w − w |) (resp. d(w, w ) = max(|w − w | ,|w − w |)). 1 2 1 T 2 1 2 1 2 If W is an open subset of C × C or of T × C and if F : W → C we set F = sup|F| (with the convention that F = 0 if W is empty). If a function W (w ,w ) → W 1 2 F(w ,w ) is differentiable enough (for the standard real differentiable structure on W) 1 2 32 k 1 k 2 1 2 we can as usual deﬁne its partial derivatives ∂ ∂ ∂ ∂ F(k , k , l , l ∈ N) and its 1 2 1 2 w w w w 1 1 2 2 With this notation D(z, 0)=∅. 32 2 Here we use the standard notation: if w = t + is, (t, s) ∈ R , ∂ = (1/2)(∂ − i∂ ) and ∂ = ∂ = (1/2)(∂ + i∂ ). w t s w w t s ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 27 l 2 j k 1 k 1 2 (total) j -th derivative D F = (∂ ∂ ∂ ∂ F) (j ∈ N). We then deﬁne k +k +l +l =j w w w w 1 2 1 2 1 1 2 2 j k l k j 1 1 2 2 D F = max ∂ ∂ ∂ ∂ F , F n = maxD F . W W C (W) W w w w w 1 1 2 2 0≤j≤n (k ,l ,k ,l )∈N 1 1 2 2 k +l +k +l =j 1 1 2 2 We denote by C (W) the set of functions F : W → C such that F n < ∞ and by C (W) O(W) the set of holomorphic functions F : W → C (all the preceding partial derivatives of the form ∂ = ∂ then vanish). w w We say that an open set W of M := C or of M := T × C is σ -symmetric (i = ∞ i 1, 2, 3) if it is invariant by σ (σ (W) = W); if W is σ -symmetric we say that a function i i i F : W → C is σ -symmetric if F ◦ σ = F (the complex conjugate of F) and we denote i i n n by C (W), resp. O (W), the set of C resp. holomorphic functions F : W → C that are σ i σ -symmetric. When no confusion is possible on the nature of the relevant σ involved, we i i shall often say σ -symmetric or even real symmetric instead of σ -symmetric. If W is σ - symmetric we use the notation W ={w ∈ W,σ(w) = w};if W = ∅ then F ∈ O (W) R R σ deﬁnes by restriction a map (still denoted by F) F : W → R. Note that a function F : (R , 0) → R which is real analytic is in O (D(0,ρ) × D(0,ρ)) for some ρ> 0. 2 n Let W be a open set of M := C or T × C.Wedenoteby Diff (W), resp. O n Diff (W), the set of C , resp. holomorphic, diffeomorphism f : W → f (W) ⊂ Mdeﬁned on an open neighborhood W of W containing the closure of W. Note that there exists a constant C depending only on M such that for any C - diffeomorphisms f , f : M → M satisfying f − id 1 ≤ 1 one has 1 2 1 C (2.43) (f ◦ f ) − id 1 ≤ C(f − id 1 +f − id 1 ). 2 1 C 1 C 2 C If now W is a σ -symmetric open set of (M,σ ) we denote by Diff (W) resp. O n O Diff (W) the set of f ∈ Diff (W), resp. f ∈ Diff (W),suchthat f ◦ σ = σ ◦ f .Itthen deﬁnes by restriction a C , resp. real analytic, diffeomorphism (that we still denote f ) f : W → f (W ) ⊂ M . R R R When f , g are two σ -symmetric holomorphic diffeomorphisms we write (2.44) [W] f = g (W possibly empty) to say that f , g ∈ Diff (W) coincideonanopenneighborhood of W containing the closure of W. 2.1. Domains W .— Let h ≥ 0 and U an open connected set of C; we shall deﬁne h,U AA AA CC CC∗ domains W of M = M = T × C (AA stands for “Action-Angle”) and W ,W h,U h,U h CC CC∗ 2 of M = M = M = C (CC for “Cartesian Coordinates”) the following way: CC∗ – Cartesian Coordinates (CC∗):if ρ := sup{|r|, r ∈ U}, the set W ⊂ C × C is h,U 2 2 x + y 1/2 CC∗ 2 h (2.45)W ={(x, y) ∈ C , |x ± iy|≤ 2e ρ , ∈ U}; h,U U 2 28 RAPHAËL KRIKORIAN R/(2π Z) AA FIG.6.—Thedomain W = T × U h,U 1/2 max(|z|,|w|) ≤ e ρ −izw ∈ U CC CC∗ 1 FIG. 7. — Schematic representation of the domain W (and W if one makes the change of coordinates z = (x + iy), h,U h,U w = (x − iy)) CC – Cartesian Coordinates (CC):if ρ := sup{|r|, r ∈ U}, the set W ⊂ C × C is h,U 1/2 1/2 CC h h (2.46)W ={(z,w) ∈ D(0, e ρ ) × D(0, e ρ ), −izw ∈ U}; h,U U U AA – Action Angle coordinates (AA): the set W of T × C is h,U AA (2.47)W = T × U. h,U 2 2 In all these three cases we denote by r the observable (x, y) → (1/2)(x + y ), (z,w) → −izw, (θ , r) → r. In Section 4.1 we shall see how one goes from (CC) (or (CC*)) to (AA) coordinates. 2.2. Cauchy estimates. — If δ> 0 wedenoteby U (W) ={w ∈ W, B(w, δ) ⊂ W} (here B(w, δ) is the ball {z ∈ M, d(z,w) < δ}). Assume that F ∈ O(W). By differentiating (k + k )- times Cauchy complex integration formula 1 2 1 F(ζ ,ζ ) 1 2 F(w ,w ) = dζ dζ 1 2 1 2 (2π i) (w − ζ )(w − ζ ) 1 1 2 2 |w −ζ |=δ |w −ζ |=δ 1 1 2 2 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 29 one sees that if U (W) is not empty k k −(k +k ) 1 2 1 2 (2.48) ∂ ∂ F ≤ C δ F . U (W) k ,k W w w δ 1 2 1 2 2.3. Holed domains. 2.3.1. Holed domain of C.— A holed domain of C is an open set of C of the form (2.49)U = D(c,ρ) D(c ,ρ ), i i i∈I for some c ∈ C, ρ> 0, c ∈ C, ρ > 0 and where I is a ﬁnite set which is either empty i i or such that for any i ∈ I, D(c ,ρ ) ∩ D(c,ρ) = ∅. Note that the disks D(c ,ρ ) are not i i i i supposed to be included in D(0,ρ). It is not difﬁcult to see that there exists a unique minimal J ⊂ I (for the inclusion) such that D(c ,ρ ) = D(c ,ρ ) and that the U i i i i i∈J i∈I representation (2.49) with I replaced by J is then unique: (2.50)U = D(c,ρ) D(c ,ρ ). i i i∈J We then denote by (2.51) D(U)={D(c ,ρ ), i ∈ J }. i i U We shall call D(c,ρ) the external disk of U. We then set ρ := rad U := ρ, rad(U) = min ρ U i∈J i 2 1/2 a(U) = ( ρ ) i∈J (2.52) card(U) = #J ⎪ U d(U) = rad(U) if J =∅, d(U) = min(rad(U), rad(U)) if J = ∅. U U If J is empty or if all the disks D(c ,ρ ), i ∈ J , are pairwise disjoint and included in U i i U D(c,ρ) we say that the holed domain U has disjoint holes and we call D(c ,ρ ) the holes i i of U (the bounded connected components of C U). We denote by D(U) the set of all these disks. Note: We shall only consider in this paper holed domains (2.49)where the c are on the real axis. 2.3.2. Holed domains of C × C or T × C.— These are by deﬁnition sets of the form W where h > 0 and U is a holed domain; see (2.46)or(2.47). We then deﬁne h,U d(W ) = min(h, d(U)). h,U 30 RAPHAËL KRIKORIAN −δ 2.3.3. Deﬂation of a holed domain. — If δ ∈ R we use the notation e D(c,ρ) for −δ −δ e D(c,ρ) = D(c, e ρ). −δ If U ⊂ C is a holed domain of the form (2.50) and if δ> 0 wedenoteby e U ⊂ Uthe (possibly empty) open set −δ −δ δ e U = D(c, e ρ) D(c , e ρ ). i i i∈J Similarly if 0 <δ < h −δ e W = W −δ . h,U h−δ/2,e U We make the following simple observations (the ﬁrst two items are proved by area considerations): Lemma 2.1. — For 1 >δ > 0 one has: (1) For any z ∈ D(c,ρ), dist(z, U) ≤ 2 a(U). 2 4δ 2 −δ (2) If ρ > 2e ρ then e U is not empty. i∈J i −δ −δ (3) If e U is not empty, then for any z ∈ e U one has D(z,(1/2)δ d(U)) ⊂ U. 2.3.4. Reformulation of Cauchy’s Inequalities. — Using item 3 of Lemma 2.1 we can in particular reformulate inequalities (2.48) when W is of the form W and F ∈ O(W ): h,U h,U m −m −m (2.53) D F −δ ≤ C δ d(W ) F . e W m h,U W h,U h,U One can sometimes obtain better estimates. – In the (AA)-case, if 0 <δ < h, one has k −k (2.54) ∂ F −δ δ F e W W θ h,U h,U −δ – In the (CC)-case, if U = D(0,ρ) and δ< 1/2 one has e W ⊂ U (W ) h,D(0,ρ) δ h,D(0,ρ) 1/2 −h with δ = ρ e δ/4and thus h −1 −1/2 −δ (2.55) ∇ F e δ ρ F . e W W h,U h,U 2.4. Whitney type extensions on domains with holes. — The discussion that follows will be useful in the construction of the KAM Normal Form of Section 7. Let U be a real symmetric holed domain (2.56)U = D(0,ρ) D(c ,ρ ), c ∈ R, i i i i∈J U ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 31 k 33 h > 0, W one of the domains deﬁned in Section 2.1 and F : W → C be a C h,U h,U σ -symmetric function i.e. F ◦ σ = F (the complex conjugate of F). We say that a C , 34 Wh 35 σ -symmetric function F : W → C is a Whitney extension for (F, W ) if h,C h,U Wh ∀ m ∈ W , F (m) = F(m). h,U j j Wh Note that since U is open this implies that for all 0 ≤ j ≤ k,D Fand D F coincide on W . h,U We shall construct such Whitney’s extensions in two situations. Lemma 2.2. — Let F ∈ O (W ). For any δ ∈]0, 1[, there exists a C , σ -symmetric func- σ h,U Wh tion F : W → C such that h,C −δ Wh (2.57) ∀ m ∈ e W , F (m) = F(m) h,U j Wh k −2k j (2.58) sup D F ≤ C(1 + #J ) (δ d(U)) maxD F . W U W h,C −δ/10 h,e U 0≤j≤k 0≤j≤k Proof. — See Section B.1 of the Appendix. Notation 2.3. — We denote by O (W ) the set of C , σ -symmetric maps F : W → C σ h,U h,C such that the restriction of F on W is holomorphic. h,U Deﬁnition 2.4. — Let A ≥ 1, B ≥ 1, U ⊂ C a σ -symmetric holed domain. We say that a σ -symmetric C function : U → C satisﬁes an (A, B)-twist condition on U if 1 1 −1 2 3 (2.59) ∀ r ∈ U ∩ R, A ≤ ∂ (r) ≤ A, and D ≤ B. 2π 2π If U is a disk D(0,ρ ) one can construct for some 0 < ρ< ρ aC , σ -symmetric 0 0 Whitney extension for on D(0, ρ) that satisﬁes an (A, B)-twist condition on D(0, ρ). Lemma 2.5. — Let ∈ O (D(0,ρ )) (ρ ≤ 2) σ 0 0 −1 2 3 (2π) (z) = ω z + b z + O(z ), ≤ 1, with b > 0. 0 2 D(0,ρ ) 2 3 Wh There exists 0 < ρ< ρ , B ≥ 0 and a C , real symmetric extension ∈ O (C) of (, D(0, ρ)) 0 σ −1 that satisﬁes an (A, B)-twist condition on C with A = 3max(b , b ). Proof. — See Appendix B.2. Notation 2.6. — We denote by TC(A, B) the set of C , real symmetric maps : C → C satisfying an (A, B)-twist condition (2.59) with U = C. Differentiability here is related to the real differentiable structure of W . h,C The exponent Wh stands for “Whitney”. See [55], [50]. 32 RAPHAËL KRIKORIAN Let U ⊂ C be a σ -symmetric connected holed domain as in (2.56). Proposition 2.7. — If ∈ O (U) ∩TC(A, B) with (2.60)8 × max(ρ , a(U)) × A × B < 1 2 −1 then the following holds. For any ν ∈]0,(6A B) [ and any β ∈ R, either for any z ∈ U −1 (2.61) |ω(z) − β|≥ν(ω = (2π) ∂) or there exists a unique c ∈] − ρ − 2Aν, ρ + 2Aν[ such that ω(c ) = β and for any z ∈ U β β D(c , 3Aν) one has |ω(z) − β| >ν. Proof. — See Appendix B.3. 2.5. Notation O .— Let h > 0, U be a holed domain, functions F ,..., F ∈ p 1 n O(W ) and l ∈ N . We deﬁne the relation h,U G = O (F ,..., F ) l 1 n as follows: there exist a ∈ N ,C > 0and Q(X ,..., X ) a homogeneous polynomial 1 n (independent of U) of degree l in the variables (X ,..., X ) such that for any 0 <δ < h/2 1 n satisfying −a −a (2.62)Cd(W ) δ maxF ≤ 1 h,U i W h,U 1≤i≤n −δ one has G ∈ O(e W ) and h,U −a −a (2.63) G −δ ≤ d(W ) δ Q(F ,...,F ). e W h,U 1 W n W h,U h,U h,U We shall use the notation O (F ,..., F ) if the polynomial Q is null when X = 0 l 1 n 1 i.e. Q(0, X ,..., X ) = 0; for example if l = n = 2, Q(X , X ) = X X + X . 2 n 1 2 1 2 When we want to keep track of the exponent a appearing in (2.62), (2.63)weshall (a) use the symbol O . When δ satisﬁes (2.62) we write a,C (2.64) δ = d (F ,..., F ; W ) 1 n h,U and we use the short hand notation (2.65) δ = d(F ,..., F ; W ) 1 n h,U to say that (2.64) holds for some positive constants a, C large enough and independent of F ,..., F ,d(W ). 1 n h,U ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 33 Remark 2.1. —Note that if U = D(0,ρ ) is a disk containing 0 and F ∈ O(W ), h,U ∗= CC, AA, one has p p/2 F(z,w) = O (z,w) ⇐⇒ ∀ 0 ≤ ρ ≤ ρ , F ρ 0 W h,D(0,ρ) p p F(θ , r) = O (r) ⇐⇒ ∀ 0 ≤ ρ ≤ ρ , F ρ 0 W h,D(0,ρ) CC AA p hence if F ,..., F ∈ O(W ) (resp. ∈ O(W ))satisfy F (z,w) = O (z,w) (resp. F = 1 n i i h,U h,U O (r)), 1 ≤ i ≤ n, one has mp−2a mp−a O (F ,..., F ) = O (z,w) (resp. O (r)). m 1 n 2.6. Summary of the various notations used in the text. – a b, a b, a b, a b, a betc. See Notation 1.1. β β – a exp(b−) means: for all β> 0, one has a exp(b − β). – A(z; λ ,λ )={w ∈ C,λ < |w − z| <λ }. 1 2 1 2 – T ={x + iy + (2π Z), x, y ∈ R, |y| < h} (i =−1). T = R/(2π Z). – ρ , d(U), a(U), J , D : see Section 2.3.1. U U U 1/2 1/2 AA CC h h –W = T × U, W ={(z,w) ∈ D(0, e ρ ) × D(0, e ρ ), −izw ∈ U}. h,U h,U U U −δ – e W = W −δ . h,U h−δ/2,e U – O (W): the set of σ -symmetric holomorphic function on W. – O (W ): see Notation 2.3. Symp (W ): see Notation 4.8. σ h,U h,U a,C a – δ = d (F; W ), δ = d(F; W ), O (F, G), O (F, G):Section 2.5. h,U h,U l – TC(A, B), (A, B)-twist condition: see Notation 2.6 and Deﬁnition 2.4. – G(f , W), L(f , W): see Notation 4.1. – = φ . For the canonical map f see (4.87)and (4.88). F F J∇ F – []· Y = Y ◦ − Y. See Section 4.7. – M (F),T F, R F: see Section 5.1. n N N –A"B = (A ∪ B) (A ∩ B). 3. A no-screening criterion on domains with holes 3.1. Harmonic measures. — Let U be a bounded open set of the complex plane with boundary ∂ U. We can deﬁne its Green function, g : U × U → R as follows: for any z ∈ U, −g(z,·) is the function equal to 0 on the boundary ∂ U of U, which is subharmonic on U, harmonic on U {z} and which behaves like log|z − w| when w ∈ Ugoes to z (this means that g(z,w) + log|z − w| stays bounded when w goes to z). The Green function g is thus nonnegative. We denote by ω : U × Bor(∂ U) →[0, 1] the harmonic measure U U of U (here Bor(∂ U) is the set of borelian subsets of ∂ U) deﬁned as follows: if z ∈ Uand I ∈ Bor(∂ U) (one can assume I is an arc for example if ∂ U is a union of circles) then the function ω (·, I) is the unique harmonic function deﬁned on U, having a continuous U 34 RAPHAËL KRIKORIAN extension to Uand such that ω (z, I) = 1if z ∈ Iand 0 if z ∈ ∂ U I. Poisson-Jensen formula (cf. [39]) asserts that for any subharmonic function u : U → C u(z) = u(w)dω (z,w) − g (z,w)u(w) U U ∂ U U where u is the usual Laplacian of u. In particular, if f is a holomorphic function on U, the application of this formula to u(z) = ln|f (z)| gives ln|f (z)|= ln|f (w)|dω (z,w) − g (z,w) U U ∂ U w:f (w)=0 and thus since g is nonnegative (3.66)ln|f (z)|≤ ln|f (w)|dω (z,w). ∂ U Though we shall not use it in this paper we mention the fact that the harmonic measure ω (z,·) can also be deﬁned in a probabilistic way by using Brownian motions: if W (t) is the value at time t of a Brownian motion issued from the point z (at time 0) and T is the stopping time adapted to the ﬁltration F of hitting I before ∂ U I, then z,I z ω (z, I) = E(1 (W (T )));hence U I z z,I (3.67)ln|f (z)|≤ E(ln|f (W (T ))|). z z,I This probabilistic interpretation is often useful in trying to get a hunch of the behavior of the harmonic measures. 3.2. Screening effect. — Assume now that |f |≤ 1onUand that |f | 1on some nonempty open subset B ⊂ U. Does this imply that f is small on “most” of U? Formula (3.66) applied to the domain U B in place of U yields for any z ∈ U B (3.68)ln|f (z)|≤ ω (z,∂ B) × lnf UB and answering the preceding question amounts to getting good estimates from below on the nonnegative function ω (z,∂ B). UB For example take U = D(0, 1) and B = D(0,σ ) (∂(U B) is then the union of the two circles of center 0 and radii σ and 1). It is easy to see that for |z|≤ 1/2 (3.69) ω (z,∂ B) = ln|z|/ ln σ ≥ ln(1/2)/ ln σ UB and the preceding formula (3.68) applied to the domain U Bshows that (3.70)lnf lnf . D(0,1/2) D(0,σ ) | ln σ| ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 35 If we assume for instance α β −N −N f ≤ e , e ≤ σ ≤ 1/10, 0 <β <α, N 1 this yields −β (3.71) ω (z,∂ D(0,σ )) N UD(0,σ ) α−β (3.72)lnf −N −1. D(0,1/2) Our aim in the next Section 3.3 will be to generalize to more general domains U, in particular to holed disks (3.73)U = D(0,ρ) D(c ,ρ ) ⊃ D(0,σ ) j j j=1 (j = 1,..., N, D(c ,ρ ) ⊂ D(0,ρ), ρ σ ), the bound from below (3.69) and its immedi- j j j ate consequence inequality (3.70). However, the spectral properties of the Laplacian (with Dirichlet boundary condi- tions for example) on a domain U obtained from removing disks from a simply connected domain ∈ R C (say the unit disk), and, in particular, the possibility of having useful estimates such as (3.71), (3.72), depend(s) on the number and the sizes of the holes of U. This fact, well known in Electrostatics under the name of screening effect,was mathemat- ically studied by Rauch and Taylor in [40] (see also [41]) where they highlight two dif- ferent regimes: on the one hand, if the sizes of the holes of U are (very) small compared with their number, the spectral properties of the Laplacian on U are very similar to that of ; on the other hand, if the holes are not so small and become dense (in some sense) in a region ⊂ (which can be of codimension 1 in ) the spectral properties of the Laplacian are similar to those of . In this latter case, the holes may act like a screen that prevents the propagation of the information “ln f −1on D(0,σ )” to the rest of the holed domain U. In Appendix C we illustrate this phenomenon on an example. We now give conditions under which this screening phenomenon is not effective. 3.3. The no-screening criterion. Proposition 3.1. — Let U be a domain U = D(0,ρ) ( D(z ,ε )), D(z ,ε ) ⊂ j j j j 1≤j≤N D(0,ρ) (ρ ∈]0, 1[)and let B ⊂ U, B = D(0,σ ). Assume that f ∈ O(U) satisﬁes f ≤ 1 and f ≤ m. ∂ B 36 RAPHAËL KRIKORIAN Then for any point z ∈ U := D(0,ρ) ( D(z , d )), 2ε < d < 1 j j j j 1≤j≤N ln(|z|/ρ) ln(d /2ρ) (3.74)ln|f (z)|≤ − ln m. ln(σ/ρ) ln(ε /ρ) j=1 Proof. — Replacing z/ρ by z, z /ρ by z , σ/ρ by σ , ε /ρ by ε and d /ρ by d ,we j j j j j j can reduce to the case ρ = 1. We then denote D = D(0, 1),D = D(z ,ε ),B = D(0,σ ). j j j By Poisson-Jensen formula (3.75) ln|f (z)|≤ ln|f (w)|dω (z,w) UB ∂(UB) ≤ ω (z,∂ B) ln m. UB We now compare ω (z,∂ B) with ω (z,∂ B). We observe that the function z → UB DB ω (z,∂ B) is the unique harmonic function deﬁned on U B which is 1 on ∂Band 0 UB on ∂ D ∪ ∂(D U); since ∂(U B) = ∂ B ∪ ∂ D ∪ ∂(D U) we deduce by the Maximum Principle that it takes its values in [0, 1]. Similarly, the function z → ω (z,∂ B) is the DB unique harmonic function deﬁned on D B which is 1 on ∂Band 0 on ∂ D, hence it takes also its values in [0, 1].So (3.76) v(·) := ω (·,∂ B) − ω (·,∂ B) UB DB is a harmonic function deﬁned on U B, −1 ≤ v ≤ 1, which is 0 on ∂ B ∪ ∂ D. For 1 ≤ j ≤ N, let v be the harmonic function deﬁned on D (B ∪ D ) which is 0 j j on ∂(D B) = ∂ D ∪ ∂Band −1on ∂ D ; by the Maximum Principle −1 ≤ v ≤ 0. j j Lemma 3.2. — The function v is harmonic on U B and on this set j=1 v ≤ v. j=1 Proof. — We notice that the function v is deﬁned and harmonic on D (B∪ j=1 N N D ) = U B. We want to compare v and v on the boundary ∂(U B) = j j j=1 j=1 ∂ D ∪ ∂ B ∪ ∂(D U).On ∂ D ∪ ∂ B the two functions v and v are equal (they are j=1 both equal to 0). To compare them on ∂(D U) we notice that ∂(D U) ⊂ ∂ D j=1 and since v =−1and for i = j , v ≤ 0 we have at each point z ∈ ∂(D U) which is j | ∂ D i N N in ∂ D , v (z)≤−1hence v ≤−1. But we have seen that −1 ≤ v ≤ 1 j i i | ∂(DU) i=1 i=1 on U B. We have thus proven that on ∂(U B) one has v ≤ v and we conclude j=1 the proof by the Maximum Principle. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 37 Because of the Maximum Principle, one has on D (B ∪ D ) ln|z − z|− ln 2 − ≤ v (z). ln ε Using Lemma 3.2 we hence get for z ∈ U, N N N ln|z − z|− ln 2 ln(d /2) j j v(z) ≥ v (z)≥− ≥− . ln ε ln ε j j j=1 j=1 j=1 On the other hand ln|z| ω (z, B) = , DB ln σ so that from (3.76) one has for z ∈ U ln|z| ln(d /2) ω (z, B) ≥ − . UB ln σ ln ε j=1 Finally since ln m ≤ 0, (3.75)gives that forany z ∈ U ln|z| ln(d /2) ln|f (z)|≤ − ln m. ln σ ln ε j=1 In particular, if for example ln(d /2) ln|z| ≤ (1/2) ln ε ln σ j=1 ln|z| then ln|f (z)|≤ (1/2) ln m, an inequality which is quite similar to (3.70). ln σ 3.4. Good triples. Deﬁnition 3.3. — Let U, U , U be three nonempty open sets of C such that, 1 2 U ⊂ U, U ⊂ U. 1 2 We say that the triple (U, U , U ) is A-good (A > 0)iffor any f ∈ O(U) such that sup |f |≤ 1, 1 2 one has lnf ≤ Alnf . U U 1 2 38 RAPHAËL KRIKORIAN FIG.8.—Atriple (U, U , U ) 1 2 Remark 3.1. — Notice that if there exists an open set U ⊂ U, U ⊂ U ,U ⊂ U 1 2 such that (U , U , U ) is A-good, then (U, U , U ) is also A-good. 1 2 1 2 Remark 3.2. — In general the fact (U, U , U ) is A-good does not imply that 1 2 (U, U , U ) is A -good with A and A comparable. For example, if U = D(0, 1),U = 2 1 1 D(1/4,σ ) U = D(3/4,σ ) with σ ,σ < 1/10, (U, U , U ) is C/| ln σ |-good while 1 2 2 1 2 1 2 2 (U, U , U ) is C/| ln σ |-good. 2 1 1 We denote by A(z; λ ,λ ),0 <λ <λ , the annulus D(z,λ ) D(z,λ ). 1 2 1 2 2 1 Here is an immediate corollary of Proposition 3.1: Corollary 3.4. — Assume that the assumptions of Proposition 3.1 hold with σ = ρ /2 −δ (b > 1). Then for all 1 ≤ i ≤ N such that d > 20ε and D(z , d ) ⊂ D(0, e ρ) (δ> 0), the i i i i triple U, A(z ; (d /10), d ), D(0,ρ /2) i i i is A-good with δ ln(d /20ρ) A = − . b| ln ρ| ln(ε /ρ) j=1 4. Symplectic diffeomorphisms on holed domains 4.1. Cartesian Coordinates (CC) and Action-Angle variables (AA). — We deﬁne on R := {(x, y), x, y ∈ R} (resp. T × R := {(θ , r), θ ∈ T, r ∈ R}) the canonical sym- CC∗ AA plectic structure (area) β := dx ∧ dy (resp. β := dθ ∧ dr). This spaceaswellas R R its symplectic structure can be complexiﬁed: the space C := {(x, y), x, y ∈ C} (resp. CC∗ T × C := {(θ , r), θ ∈ T , r ∈ C}) carries the symplectic structure β := dx∧ dy (resp. ∞ ∞ C ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 39 AA 2 CC∗ β := dθ ∧ dr) and the involution σ (resp. σ )deﬁnedin(2.42) preserves (C ,β ) 1 3 C C AA 2 CC∗ AA (resp. (T × C,β ))and ﬁxes (R ,β ) (resp. (T × R,β )). C R R When working in the elliptic ﬁxed point case, it will be more convenient to use other Cartesian coordinates. Let’s introduce the (holomorphic) complex change of coor- 2 2 dinates ϕ : C → C , ϕ : (x, y) → (z,w), 1 1 √ √ z = (x + iy) x = (z − iw) 2 2 (4.77) ⇐⇒ i −i √ √ w = (x − iy) y = (z + iw). 2 2 We see that (σ is as in (2.42)) with the notations of Section 2.1 ∗ −1 CC∗ CC dx ∧ dy = ϕ (dz ∧ dw), ϕ ◦ σ ◦ ϕ = σ,ϕ(W ) = W . 1 2 h,U h,U CC CC CC∗ CC We shall denote (M ,β ,σ ), resp. (M ,β ,σ ), (CC stands for Carte- 2 1 2 CC sian Coordinates) the space C endowed with the symplectic structure β := dz ∧ dw, CC∗ AA AA resp. β = dx ∧ dy, and the involution σ , resp. σ . Similarly, (M ,β ,σ ) (AA for 2 1 3 Action-Angle coordinates) is the space T × C endowed with the symplectic structure AA β := dθ ∧ dr and the involution σ . We shall use for short the generic notation (M,β,σ ) to denote either of the preceding sets endowed with their symplectic structure and invo- lution. We also use the notation M or (M) for M ∩ σ(M).The 2-form β restricted R R to M is still a symplectic form. We shall call the origin OinM , the set O ={(0, 0)} if R R 2 CC CC AA M = C = M or M and O = T×{0} if M = M = T × C. If W is a nonempty open set of M (resp. M ) we say that f ∈ Diff (W) (resp. C ∗ f ∈ Diff (W))is symplectic if it preserves the canonical symplectic form β : f β = β . O C We denote by Symp (W) (resp. Symp (W)) the set of such symplectic holomorphic 1 O (resp. C ) diffeomorphisms. If furthermore f ◦ σ = σ ◦ f we write f ∈ Symp (W). We shall say that a symplectic diffeomorphism f is exact symplectic if there exists a 1- form λ,the Liouville form,suchthat dλ = β and f λ − λ is exact: there exists a function Ssuch that f λ − λ = d S; S is called the generating function of f (w.r.t. λ). We then denote O C f ∈ Symp (W) (resp. f ∈ Symp (W)). In our case the relevant Liouville forms will be ex.,σ ex. (4.78) (AA) λ = rdθ, (CC) λ = (1/2)(wdz− zdw), (CC*) λ = (1/2)(xdy− ydx). Let W ⊂ Mbe σ -symmetric (σ(W) = W) and such that (W) := W ∩ σ(W) = W ∩ M R R is a nonempty open set of M . Then, if f ∈ Symp (W), its restriction f : M ⊃ R |(W) R ex.,σ (W) → f ((W) ) ⊂ M deﬁnes a real analytic (exact) symplectic diffeomorphism. If R R R S ⊂ Wis f -invariant (f (S) = S) the set (S) := S ∩ M is also left invariant by f . R R |(W) Notice that if U ⊂ C is a real symmetric open set such that U ∩ R = ∅ we have AA AA ⎪ (W ) = (T × U) = T × (U ∩ R) = W R h R ⎨ h,U 0,U∩R CC CC CC (W ) ={(z,w) ∈ W ,w = iz}= (W ) R R h,U 0,U∩R 0,U∩R + + 2 2 x +y CC∗ 2 CC∗ (W ) ={(x, y) ∈ R , ∈ U ∩ R }= W . R + h,U 0,U∩R 2 + 40 RAPHAËL KRIKORIAN In any case (W ) ={r ∈ U}∩ M ={r ∈ U ∩ R}∩ M . h,U R R R There are symplectic changes of coordinates ψ that allow to pass from the (z,w)- coordinates ((CC)-coordinates) to the (θ , r)-coordinates ((AA)-coordinates). They are de- 1/2 is 1/2 is/2 ﬁned as follows. The maps r → r , te → t e for t > 0and −π< s <π (resp. for t > 0and 0 < s < 2π ) deﬁne holomorphic functions on C R (resp. on C R ). We − + can thus deﬁne the biholomorphic diffeomorphisms T × (C R ) (θ , r) −→ (z,w)∈{(z,w) ∈ C , −izw/∈ R } ∞ ± ± (4.79) iπ/4 1/2 −iθ r =−izw z = e r e ⇐⇒ 1/2 (−izw) iπ/4 1/2 iθ iθ −iπ/4 w iπ/4 w = e r e e = e = e 1/2 (−izw) z which satisfy −1 dz ∧ dw = dθ ∧ dr and ψ ◦ σ ◦ ψ = σ . ± 2 3 Notice that if h > 0 ψ −izw ∈ D(0,ρ) R (4.80) T × (D(0,ρ) R ) −→ (z,w) ∈ C , h ± −2h 2h e < |z/w| < e hence with the notations of Section 2.1 CC AA (4.81)W ⊃ ψ (W ). h,UR h,UR ± ± CC AA 4.2. Symplectic vector ﬁelds. — If (M,β) = (M ,β) or (M ,β) and F ∈ O (M) we deﬁne the holomorphic symplectic vector ﬁeld X by i β = d F. If J is the matrix one has −10 X = J∇ F. t 1 We denote by φ the ﬂow at time t ∈ R of the vector ﬁeld J∇Fand = φ its J∇ F J∇ F time 1-map. It is a symplectic diffeomorphism. If G : M → R or C is another smooth observable we deﬁne the Poisson bracket of F and G by the formula {F, G}= β(X , X ) or equivalently F G {F, G}:=∇ F, J∇ G. One then has (G ◦ ) = L G={F, G}, [L , L ]= L . |t=0 J∇ F X X X F F G {F,G} dt If f is a symplectic diffeomorphism one has −1 −1 ∗ −1 (4.82) f ◦ ◦ f = , where f F = (f ) F = F ◦ f . F f F ∗ ∗ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 41 2 2 4.3. Integrable models. — We assume that (M,β,σ ) is (C , dx ∧ dy,σ ), (C , dz ∧ dw, σ ) or (T × C, dθ ∧ dr,σ ). In all these examples there exists a natural (Lagrangian) 2 ∞ 3 foliation given by the level lines of the observable r : M → C 2 2 x + y (4.83) r(x, y) = , r(z,w)=−izw, r(θ , r) = r, which has the property that for every m ∈ M, such that r(m) ∈ R,the map R t → φ (m) is 2π -periodic. In particular, for c ∈ R, the set {r = c}⊂ M is itself foliated by the J∇ r t 1 2π -periodic orbits of the ﬂow φ ; they are either points or homeomorphic to S .We J∇ r shall say that a symplectic diffeomorphism of M is integrable if it is symplectically conju- gated to a diffeomorphism that leaves globally invariant each level line of the preceding function r. It is not difﬁcult to see that a diffeomorphism satisfying the previous condition is of the form where H = ◦ r. Let U be a σ -symmetric holed domain of C and ∈ O (U). Then, −i∂(r) i∂(r) (CC) : (z,w) = (e z, e w), (4.84) (AA) : (θ , r) = (θ + ∂(r), r), −i∂(r) −i∂(r) (CC∗) : (x, y) = (#(e (x + iy)),$(e (x + iy))) and in any case (W ) ⊂ W , h = h−$(∂ ) . h,U U h,U On the other hand, since is σ -symmetric, one has whenever U is σ -symmetric, ((W ) ) = ((W ) ). h,U R h,U R Notice that in all cases is an integrable diffeomorphism of M. CC∗ 2 CC AA 4.4. KAM circles. — A circle of M (M equals M = R ,M ,M = T× R)is R R R R R any set of the form ({r = c}) = ({r = c)}∩ M , c ∈ R, of cardinal > 1(r is the observable R R of (4.83)). In the (AA) resp. (CC*) cases this set coincides with the usual circle T×{r = c} 2 2 2 resp. {(x, y) ∈ R ,(1/2)(x + y ) = r}; in the (CC) or (CC*) cases ({r = c}) is a circle if and only if c > 0 (it is empty if c < 0 and reduced to {(0, 0} if c = 0). C 1 Let W be an open subset of M and f ∈ Symp (W) aC symplectic diffeo- ex. morphism W → f (W). For example f could be the restriction on W = (W ) of h,U R f ∈ Symp (W ),W ⊂ M. A KAM-circle (or KAM-curve)for f is the image g(({r = h,U h,U ex.,σ c}) ) ⊂ Wof a circle ({r = c}) , c ∈ R,bya C symplectic diffeomorphism g : M → M R R R R ﬁxing the origin (g({r = 0} )={r = 0} )and such that R R −1 g ◦ f ◦ g = + O(r − c), ω ∈ R Q. 2πωr 42 RAPHAËL KRIKORIAN The set g(({r = c}) ) ⊂ Wis then f -invariant, homeomorphic to S and non homotopi- cally trivial in the following sense: in the (AA)-case it is homotopic to {r = 0} = T×{0} and in the (CC) or (CC*) case it has degree ±1 w.r.t. to the origin {r = 0} ={(0, 0)}. Moreover, the restriction of f on g(({r = c}) ) ⊂ W is conjugated to a rotation on a circle with frequency ω ∈ R. Notation 4.1. — We denote by G(f , W) the set of f -invariant KAM-circles γ ⊂ (W) and by L(f , W) ⊂ (W) their union: L(f , W) = γ . γ∈G (f ,W) Remark 4.1. —Let g, f , f , f : M → M be C symplectic diffeomorphisms 1 2 R R where g({r = 0} )={r = 0} . Then, R R (1) If A ⊂ B ⊂ (M) ,then L(f , A) ⊂ L(f , B). (2) If f , f coincide on a set A, L(f , A) = L(f , A). 1 2 1 2 (3) For any set A ⊂ M −1 (4.85) g(L(f , A)) = L(g ◦ f ◦ g , g(A)). −1 (4) If g ◦ f ◦ g and f coincide on a set A one has 1 2 (4.86) L(f , g(A)) = g(L(f , A)). 1 2 Deﬁnition 4.2. — If A ⊂ C we deﬁne W ={r ∈ A}∩ M ={r ∈ A ∩ R}∩ M . A R R Let us now state a criterion that ensures the existence of KAM-circles. Assume that there exist ∅ = L ⊂ A = I I ⊂ A ⊂ R, j∈J where L is compact and A is of the form I I where I ⊂ R is an interval and the I j j j∈J are pairwise disjoint intervals. C 1 Proposition 4.3. — Let f ∈ Symp (W ) and suppose that there exist ∈ C (R) and a 1 −1 C symplectic diffeomorphism g : M → M ﬁxing the origin, g − id 1 ≤ C (C depends only R R C on M), such that −1 on W g ◦ f ◦ g = and g(W ) ⊂ W . L (r) L A 1/2 Then, if |I | ≤ 1, one has j∈J 1/2 Leb (W L(f , W )) ≤ C × (Leb (A L)+g − id ). M A R 0 R A Proof. — Since W = L( , W ) one has from (4.86) g(W )= g(L( , W ) = L (r) L L (r) L L(f , g(W )) and since g(W ) ⊂ W one has g(W ) ⊂ L(f , W ). On the other hand L L A L A ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 43 if we deﬁne E by W = W ∪ E, one has g(W ) = g(W ) ∪ g(E) and thus g(W ) ⊂ A L A L A L(f , W ) ∪ g(E). We therefore have Leb (W L(f , W )) Leb (g(E)) + Leb (W " g(W )). M A A M M A A R R R 1/2 Since A = I I and |I | ≤ 1, Lemma J.1 from the Appendix yields j j j∈J j∈J 1/2 Leb (W " g(W ))≤g − id and since Leb (g(E)) = Leb (E) we get the con- M A A 0 M M R R R clusion. 4.5. Generating functions. — Let h > 0, U ⊂ C be a real symmetric holed domain AA CC and W and W the domains deﬁned in (2.47)and (2.46) h,U h,U AA W = T × U h,U 1/2 1/2 CC h h W ={(z,w) ∈ D(0, e ρ ) × D(0, e ρ ), r := −izw ∈ U}. h,U U U We shall associate to each F ∈ O (W ) small enough a real symmetric holomorphic σ h,U symplectic diffeomorphism f of W which is exact with respect to the respective Liouville F h,U forms as deﬁned in (4.78)). It is deﬁned as follows: in the (AA)-case ϕ = θ + ∂ F(θ , R) (4.87) f (θ , r) = (ϕ, R) ⇐⇒ r = R + ∂ F(θ , R) and in the (CC)-case z = z + ∂ F(z, w) (4.88) f (z,w) = ( z, w) ⇐⇒ w = w + ∂ F(z, w) . Lemma 4.4. — There exists a constant C such that if F ∈ O (W ) and 0 <δ < h satisfy σ h,U −2 (4.89) C(δ d(W )) F < 1, h,U W h,U the map f deﬁned by (4.87), (4.88) is a real symmetric holomorphic exact symplectic diffeomorphism −δ from e W onto its image and h,U −2δ −δ (4.90) e W ⊂ f (e W ) ⊂ W . h,U h,U h,U We shall call f the generating map of F.Moreover −1 (4.91) f = f . −F+O(D FDF) Proof. — See Appendix A.1. 44 RAPHAËL KRIKORIAN Remark 4.2. — The symplectic change of coordinates ψ introduced in Sec- CC tion 4.1 preserves exact symplecticity: if f is exact symplectic the same is true for AA −1 CC iπ/4 1/2 −iθ iπ/4 1/2 iθ f = ψ ◦ f ◦ ψ . Indeed, if ψ (θ , r) = (z,w), z = e r e , w = e r e ,one ± ± computes the Liouville form (1/2)(wdz − zdw) = rdθ . Conversely, if a diffeomorphism (θ , r) → (ϕ, R) is exact symplectic and close enough to the identity, it admits this type of parametrization. More precisely: Lemma 4.5. — Let f ∈ Symp (W ) be an exact symplectic diffeomorphism close enough h,U ex.σ −δ to the identity. Then, if δ = d(f − id, W ) (recall the notation (2.64)) there exists F ∈ O (e W ) h,U σ h,U −δ such that on e W one has h,U f = f , F = O(Df − id) = O (f − id). F 1 This F is unique up to the addition of a constant. Conversely, given F ∈ O(W ) one has h,U (4.92) f = ◦ f = id + J∇ F + O(D FDF). F F O (F) Proof. — See Appendix A.2. The composition of two exact symplectic maps is again exact symplectic and more precisely −δ Lemma 4.6. — Let F, G ∈ O(W ).If δ = d(F, G; W ) then on e W , h,U h,U h,U (4.93) f ◦ f = f F G F+G+O(DF DG ) h,U h,U (4.94) f = f ◦ f = f ◦ f . F+G F+DF O (G) G F G+DG O (F) h,U 1 h,U 1 In the Action-Angle case, if depends only on the variable r then = f and (4.95) ◦ f = f F +F Proof. — See the Appendix, Section A.3. 4.6. Parametrization. — We shall parametrize perturbations of integrable symplectic diffeomorphisms deﬁned on a domain W by h,U f = ◦ f (r) F where ∈ O (U) and F ∈ O (W ).Notethatif f = id + O (z,w) or f (θ , r) = id + σ σ h,U F (O(r), O(r )) then: 3 2 Case (CC) F(z,w) = O (z,w), Case (AA) F(θ , r) = O(r ). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 45 4.7. Transformation by conjugation. — We now deﬁne (4.96) []· Y = Y ◦ − Y. Note that (AA)-case if Y = Y(θ , r), ([]· Y)(θ , r) = Y(θ + ∂(r), r) − Y(θ , r); (4.97) (CC)-case if Y = Y(z,w), −i∂(r) i∂(r) ([]· Y)(z,w) = Y(e z, e w) − Y(z,w). If W = W is a holed domain and δ> 0 we introduce the notation h,U W = W := W ∪ (W ). h,U h,U h,U h,U The main result of this section is the following: Proposition 4.7. — Let ∈ O (U), F ∈ O (W ), Y ∈ O (W ).Then, if δ = σ σ h,U σ h,U −δ d(F, W ) ∩ d(Y, W ) there exists F ∈ O (e W ) such that h,U σ h,U h,U −δ −1 [e W ] f ◦ ( ◦ f ) ◦ f = ◦ f h,U Y F F (see the notation (2.44)) and F = F+[]· Y+DF O (Y) W 1 = F+[]· Y + O (Y, F). Proof. — See the Appendix, Section A.4. Remark 4.3. — A direct computation shows that if (r) = 2πω r + O(r ) and k k Case (CC) F(z,w) = O (z,w) and Y(z,w) = O (z,w), k k Case (AA) F(θ , r) = O(r ) and Y(θ , r) = O(r ) then 2k−2 2k−1 Case (CC) F(z,w) = O (z,w), Case (AA) F(θ , r) = O (r). 4.8. Symplectic Whitney extensions. — Let U ⊂ C be a real symmetric holed domain Wh 2 W ⊂ M, F ∈ O (W ) and F : M → C be a σ -symmetric C Whitney extension of h,U σ h,U (F, W ) (cf. Section 2.4). There exists a constant C > 0 (depending only on M) such that h,U Wh −1 1 2 Wh if F < C , Equations (4.87), (4.88)deﬁne a C -diffeomorphism f : M → M C (M) F such that −1 −1 Wh (4.98)max(f Wh − id 1 ,f − id 1 ) ≤ C F 2 . F C (M) Wh C (M) C (M) F 46 RAPHAËL KRIKORIAN −1 −1 1 −δ −δ Note that f Wh and f are C σ -symmetric extensions of (f , e W ) and (f , e W ) F Wh F h,U h,U for any δ satisfying (4.89), cf. Lemma 4.4. Wh Wh In general, the diffeomorphism f is not symplectic on M but since F takes real values on M , f Wh : M → M is an exact symplectic diffeomorphism of M . R F R R R Notation 4.8. — We shall denote by Symp (W ),resp. Symp (W ), the set of C h,U h,U σ ex.,σ O O σ -symmetric diffeomorphisms M → M that are in Symp (W ),resp. Symp (W ), (hence h,U h,U σ ex,s holomorphic on W ) and symplectic, resp. exact symplectic, when restricted to M → M . h,U R R 5. Cohomological equations and conjugations Our aim in this section is to provide a uniﬁed treatment, both in the (AA) and (CC) cases, of the resolution of the (co)homological equations (Proposition 5.3)involvedinthe Fundamental conjugation step (Proposition 5.5) that we shall use to construct all our dif- ferent Normal Forms (for instance the approximate Birkhoff Normal Form of Section 6, the KAM Normal Forms of Section 7 and the resonant Normal Form of Appendix G). 5.1. Fourier coefﬁcients and their generalization. — In this section we assume that either: – Case (CC): (M,β) = (C × C, dz ∧ dw) and we denote by r(z,w)=−izw –or, Case (AA): (M,β) = (T × C, dθ ∧ dr) and we denote by r : (θ , r) → r. In both cases the ﬂow t → φ is 2π -periodic w.r.t. t ∈ R (cf. (4.84)). J∇ r Let U be a connected open set of C and F ∈ O(W ).For any m ∈ W and any h,U h,U t ∈ R, φ (m) ∈ W : h,U J∇ r t −it it (CC) : φ (z,w) = (e z, e w), J∇ r (AA) : φ (θ , r) = (θ + t, r). J∇ r We can hence deﬁne t → F(φ (m)) which is a 2π -periodic function R → C and for J∇ r n ∈ Z we introduce its n-th Fourier coefﬁcient M (F)(m): 2π −int t (5.99) M (F)(m) = e F ◦ φ (m)dt J∇ r 2π t int (5.100) F(φ (m)) = M (F)(m)e . J∇ r n∈Z Thedependenceof M (F)(m) is holomorphic in m andwehavethusdeﬁned M (F) ∈ n n O(W ). We observe that h,U 2π/n (5.101) M (F) ◦ φ = M (F) n n J∇ r and ∀ t ∈ R, M (F) ◦ φ = M (F). 0 0 J∇ r ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 47 5.1.1. Case (CC). — One has t −it it (5.102) φ (z,w) = (e z, e w) J∇ r and if F = F(z,w),(5.99) becomes 2π −int −it it M (F)(z,w) = e F(e z, e w)dt. 2π k l If furthermore F(z,w) = F z w is converging on some polydisk D(0,μ) × k,l (k,l)∈N D(0,μ) one has k l (5.103) M (F)(z,w) = F z w n k,l (k,l)∈N l−k=n ∗ p hence, if for some p ∈ N ,F(z,w) = O (z,w),thenfor any n ∈ N, M (F)(z,w) = O (z,w). 5.1.2. (AA) Case. — In that case φ (θ , r) = (θ + t, r) J∇ r and if F = F(θ , r) we deﬁne 2π −1 −int M (F)(θ , r) = (2π) e F(θ + t, r)dt inθ = F(n, r)e where 2π −1 −inθ F(n, r) = (2π) e F(θ , r)dθ is the n-th Fourier coefﬁcient of F(·, r). Notice that though F is only deﬁned on T × U, M (F) is deﬁned in T × U. n ∞ Remark 5.1. — We see from (5.99)thatiffor some p > 0, F = O (r) (which means p p that for any m ∈ W one has |F(m)|≤ C|r(m)| for some C > 0) then M F = O (r) for h,U n any n ∈ N. 2π/n Remark 5.2. — Using the fact that M (F) ◦ φ = M (F) one can show that n n J∇ r 2π/n 2π/n f ◦ φ = φ ◦ f both in the (AA) and (CC) Case. F F J∇ r J∇ r 2π/n 36 −2π i/n 2π i/n For example in the (CC)-case, since φ (z,w) = (e z, e w), the condition on F implies J∇ r −2π i/n 2π i/n F(e z, e w) = F(z, w) and the conclusion follows from (4.88). 48 RAPHAËL KRIKORIAN 5.1.3. For m of M (F). Lemma 5.1. — If F ∈ O (W ) there exists M(F) ∈ O (U) such that σ h,U σ (5.104) M (F) = M(F) ◦ r, M(F) ≤F . 0 U h,U Moreover (5.105) f = ◦ f . M(F) M(F) O (F) Proof. — By deﬁnition of M (F) we see that for every t ∈ R M (F) ◦ φ = M (F). 0 0 J∇ r Lemma D.1 of the Appendix provides us with M(F) ∈ O (U) such that M (F) = σ 0 M(F) ◦ r.Wejusthavetoprove (5.105) in the (CC) case. If ( z, w) = f (z,w) one has M(F) −1 z = (1 + ∂(M(F))(zw) )z, w = (1 + w∂ (M(F))(zw) ) w and since w( z,w) = w + O(F) we get −∂(M(F))(zw) ∂(M(F))(zw) ( z, w) = (e z, e w) + O (F). 5.1.4. Decay of the M (F).— We observe that –in Case (AA),for m = (θ , r) ﬁxed in W , the function h,U T → C h−|$θ| t → F(φ (m)) = F(θ + t, r) J∇ r is well deﬁned and holomorphic; CC –in Case (CC),for (z,w) ∈ W ﬁxed (recall (5.102) and the deﬁnition (2.46)of W ), h,U h,U the function h 1/2 h 1/2 R + i]− ln(e ρ /|w|), ln(e ρ /|z|)[→ C (5.106) t −it it t → F(φ (m)) = F(e z, e w) J∇ r (with ρ = sup{|r|, r ∈ U}) is also a well deﬁned 2π Z-periodic holomorphic func- CC h−δ 1/2 tion. Furthermore, if m = (z,w) ∈ W one has max(|z|,|w|) ≤ e ρ thus h−δ,U h 1/2 h 1/2 min(ln(e ρ /|w|), ln(e ρ /|z|)) ≥ δ. Hence, in any case, for m ∈ W the function t → F◦ φ (m) is 2π -periodic, holomor- h−δ,U J∇ r phic on T and bounded in module by F . The Fourier coefﬁcients M (F)(m) of δ W n h,U the function t → F ◦ φ (m) J∇ r t int (5.107)F ◦ φ = e M (F) J∇ r n∈Z ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 49 thus satisfy 1/2 (5.108) |M (F)(m)| ≤F n W h,U n∈Z and in fact (cf. for example [44]) 1/2 2|n|δ 2 1/2 (5.109) e |M (F)(m)| ≤ 2 F . n W h,U n∈Z 5.1.5. Truncations operators. — Let us deﬁne for N ∈ N∪{∞} , T F = M (F), R F = F − T F. N n N N |n|<N Lemma 5.2. — If F ∈ O(W ) one has h,U (5.110) on W , F = M (F) h,U n n∈Z −|n|δ (5.111) M (F) e F , n W W h−δ,U h,U −1 −Nδ (5.112) R F δ e F , N W W h−δ,U h,U −1 −Nδ (5.113) T F F (if δ e ≤ 1). N W W h−δ,U h,U Furthermore, if for some p > 0, F = O (r) then (5.114)R F = O (r); in the (CC Case), if F ∈ O(W ) ∩ O (z,w), one has h,U (5.115) (R F)(z,w) = O (z,w). Proof. — Inequality (5.111) is a straightforward consequence of (5.109). Equality (5.110) comes from taking t = 0in (5.100). (5.112) is a consequence of (5.111)and (5.113) is clear from (5.112). Inequalities (5.114)and (5.115) are consequences respectively of Remark 5.1 and of identity (5.103). 5.2. Solution of the truncated cohomological equation. — We assume that 0 <ρ ≤ 1and that U is a σ -symmetric open connected set of D. We recall that we have deﬁned in (4.96)(cf. Proposition 4.7)for any ∈ O(U) and Y ∈ O(W ) h,U []· Y = Y ◦ − Y. (r) The main Proposition is the following: 50 RAPHAËL KRIKORIAN Proposition 5.3. — Let τ ≥ 0, ∈ O (U), K > 0, N ∈ N ∪{∞} be such that one has on U ∗ −1 −τ (5.116) ∀ (k, l) ∈ N × Z, 1 ≤ k < N %⇒ |k ∂(·) − l|≥ K |k| . 2π Then, for any F ∈ O (W ), there exists Y ∈ O (W ) such that, on W , one has M (Y) = 0, σ h,U σ h,U 0 h,U M (Y) = 0 for |k|≥ N and (5.117)T F − M (F)=[]· Y. N 0 This Y satisﬁes for any 0 <δ < h −(1+τ) τ+1 (5.118) Y Kmin(δ , N )F . h,U h−δ,U Moreover, if we assume in addition that is of the form (r) = 2πω r, ω ∈ R,thenone canimprove 0 0 the exponent in (5.118): −τ τ (5.119) Y Kmin(δ , N )F . h,U h−δ,U Proof. — We observe that both in Case (AA) or Case (CC) one has on W ∩{r ∈ R} h,U (cf. (4.84)) ∂(r) = φ . (r) J∇ r Hence, if G is a function in O(W ) one has on W ∩{r ∈ R} h,U h,U 2π t+∂(r) −int M (G) ◦ = e G ◦ φ dt n (r) J∇ r 2π 2π in∂(r) −int t = e e G ◦ φ dt J∇ r 2π in∂(r) = e M (G) and since M (G) ∈ O(W ), the left hand side of the preceding equations can be holo- n h,U morphically extended to a function in O(W ).Wethenhavein O(W ) h,U h,U in∂(r) []· M (G) = (e − 1)M (G). n n in∂(r) −1 −τ Note that from Lemma M.1 one has for r ∈ U, |e − 1|≥ K |n| .Ifwedeﬁne Y by (5.120)Y = M (F) in∂(r) e − 1 0<|n|<N ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 51 we have from Lemma 5.2 τ −|n|δ Y K |n| e F W h,U h−δ,U 1≤|n|<N −(1+τ) τ+1 min(Kδ , KN )F h,U and []· Y = Y ◦ − Y = T F − M (F). N 0 This last formula shows that if we deﬁne Yon (W ) by Y ◦ = T F − (r) h−δ,U (r) N M (F) + Y the functions Y and Y coincide on (W ) ∩ W and thus Y can 0 (r) h−δ,U h−δ,U be holomorphically extended to (W ) ∪ W =: W and (r) h−δ,U h−δ,U h−δ,U −(1+τ) τ+1 Y min(Kδ , KN )F , h,U h−δ,U which is (5.118). The fact that M (Y) = 0 and its uniqueness (under the condition M (Y) = 0) 0 0 comes again from Lemma 5.2. Finally, the σ -symmetry of Y on W is clear. h,U −1 To conclude the proof of the Proposition we just have to check that if (2π) (r) ≡ ω ∈ R satisﬁes (5.116)then(5.119) holds. This is a result due to Rüssmann [45]thatwe now recall for completeness. In fact, Rüssmann proves that if D = min|nω − l|, D = min D n 0 j l∈Z 1≤j≤n one has −2 2 ∗ −2 (5.121) D ≤ (π /3)(D ) . n N n=1 2π inω From Lemma M.1 one has |e − 1|≥ 4min |nω − l|= 4D . Thus, if we apply l∈Z 0 n Cauchy-Schwarz inequality to (5.120)wehavefor ν = 0or ν = δ 1/2 1/2 −2|n|ν −2 2|n|ν 2 Y e D e M (F) W n h−δ,U n W h−δ,U 1≤|n|<N 1≤|n|<N 1/2 −2|n|ν −2 e D F (cf. (5.108), (5.109)). n h,U 1≤|n|<N –If N < ∞,wetake ν = 0and (5.121)gives ∗ −1 Y (D ) F W h,U h−δ,U N KN F . h,U 52 RAPHAËL KRIKORIAN –If N =∞ we take ν = δ. Taking into account (5.121), we perform an Abel summa- −2|n|ν −2 −2|n|ν −2 tion (discrete integration by part) on the sums e D , e D : 1≤n<N n 1≤−n<N n this yields 1/2 −2|n|δ −2(|n|+1)δ ∗ −2 Y (e − e (D ) F W h,U h−δ,U n 1≤|n|<∞ 1/2 −2|n|δ 2 2τ δe K |n| F h,U 1≤|n|<∞ −τ Kδ F . h,U Remark 5.3. —If in Proposition 5.3 U = D(0,ρ) is a disk centered at 0 and ilθ k (AA)-case F(θ , r) = F (l)e r ⎪ k k∈N l∈Z k l (CC)-case F(z,w) = F z w ⎪ k,l (k,l)∈N one has the more explicit expressions F (l) ilθ k (AA)-case Y(θ , r) = e r il∂(r) ⎨ e − 1 k∈N l∈Z (5.122) k,l k l (CC)-case Y(z,w) = z w . ⎪ i(l−k)∂ (r) ⎪ e − 1 (k,l)∈N l =k In particular, if (CC)-case F(z,w) = O (z,w) (r) = 2πω r and (AA)-case F(θ , r) = O (r) then Y satisﬁes also (see the remarks at the end of Sections 5.1.1 and 5.1.2) (CC)-case Y(z,w) = O (z,w) (AA)-case Y(θ , r) = O (r). 5.3. Fundamental conjugation step. — We begin by the following consequence of Proposition 4.7. Let U be a holed domain, h > 0. Lemma 5.4. — There exists a ≥ 2 and C > 0 such that if ∈ O (U), F ∈ O (W ), σ σ h,U Y ∈ O (W ) and δ = d(F, Y; W ), δ> 0 satisﬁes σ h,U h,U −a −a (5.123)C d(W ) δ F ≤ 1 h,U W h,U ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 53 −δ then one has on e W (cf. Lemma 5.1 for the deﬁnition of M(F)) h,U −1 (5.124) f ◦ ◦ f ◦ f = ◦ f (a) . Y F +M(F) ˙ F−M (F)+[+M(F)]·Y+O (Y,F) Proof. — We ﬁrst observe that since F = F − M (F) + M (F),wehaveby(4.94) 0 0 and Lemma 5.1 f = f ◦ f F M (F) F−M (F)+O (F) 0 0 2 = ◦ f ◦ f M(F) O (F) F−M (F)+O (F) 2 0 2 = ◦ f M(F) F−M (F)+O (F) 0 2 and thus ◦ f = ◦ f . F +M(F) F−M (F)+O (F) 0 2 Now we use Proposition 4.7 and make explicit the notations d and O:for some a > 0that we can choose ≥ 2and some C > 0, if (5.123)issatisﬁed, onehas −1 (5.125) f ◦ ◦ f ◦ f Y +M(F) F−M (F)+O (F) 0 2 Y (a) = ◦ f . +M(F) F−M (F)+[+M(F)]·Y+O (Y,F) Proposition 5.5. — Let a = a + 4 (a from Lemma 5.4). There exists C > 0 such that the following holds. Let U be a holed domain, ∈ O (U),and F ∈ O (W ). Assume that there exists σ σ h,U a holed domain V ⊂ U, N ∈ N ∪{∞} and K > 0 such that on V the following non-resonance condition (cf. (5.116)) is satisﬁed: ∗ −1 −τ (5.126) ∀ (k, l) ∈ N × Z, 1 ≤ k < N %⇒ |k ∂(·) − l|≥ K |k| 2π −1 −1 −δ and assume that CN <δ < min(h, C ) is such that e W is not empty and h,V −1 −(a +τ) (5.127) (δ d(W )) KF < C . h,V h,U Then there exists Y ∈ O(W ) solution on W of the cohomological equation (cf. (5.117), (5.118)): h,V h,V −(1+τ) (5.128)T F − M (F)=−[]· Y, Y −δ/2 Kδ F N 0 W e W h,U h,V −δ −δ and ∈ O(e W ), F ∈ O (W ) such that one has on e W h,V σ h,U h,V −1 f ◦ ◦ f ◦ f = ◦ f , = + M(F) Y (r) F (r) F (5.129) −(a +τ) 2 −Nδ/2 F 3 −δ ≤ K(δ d(W )) F + e F . C (e W ) h,V h,U h,V h,U 54 RAPHAËL KRIKORIAN Proof. — We apply Proposition 5.3 to obtain some Y satisfying (5.128)and we apply Lemma 5.4 with δ equal to δ/2. Since (cf. (4.97)) [ + M(F)]· Y =[]· Y + (2) O(|∇ Y||∇(M(F))|)=[]· Y+ O (Y, F), we get using []· Y+ F− M (F) = R F(cf. 0 N (5.117)), −δ/2 −1 e W , f ◦ ◦ f ◦ f =: ◦ f h,V Y F with (5.130) = + M(F) (a) (5.131) F = R F + O (Y, F). (a) The deﬁnition of the symbol O ,(5.112)and (5.128) show that there exists a universal positive constant Csuch that if (5.127) is satisﬁed one has −(1+τ) −1 −1 a 2 −1 −Nδ/2 (5.132) F −δ/2 Kδ (δ d(W ) ) F + δ e F . e W h,V h,U h,V h,U Inequalities (5.129), comes from (5.132) and Cauchy’s inequality (2.53)ofSection 2.3.4 −δ −δ/2 −δ −δ/2 (applied with e W and e W in place of e W and W ) because d(e W ) ≥ h,V h,V h,U h,U h,V (1/2) d(W ) if δ< 1/10 (which is the case if C is large enough). h,V 6. Birkhoff Normal Forms 6.1. Formal Normal Forms. — We recall in this subsection the classical results on (formal) Birkhoff Normal Forms. For more details on the related formal aspects we refer to Appendix E. We also explain how Pérez-Marco’s dichotomy extends to the diffeomor- phism case (in particular in the (AA)-case). 6.1.1. BNF near a non-resonant elliptic ﬁxed point ((CC) case). — Let f : (R , 0) → (R , 0) be a real analytic symplectic diffeomorphism of the form f (x, y) = Df (0, 0) · (x, y) + O (x, y) where cos(2πω ) − sin(2πω ) 0 0 Df (0, 0) = = 2πω r sin(2πω ) cos(2πω ) 0 0 with ω ∈ R Q. 2 2 If ϕ : C → C is the change of coordinates ϕ(x, y) = (z,w) deﬁned in (4.77) −1 the diffeomorphism f := ϕ ◦ f ◦ ϕ is exact symplectic and of the form f (z,w) = (z,w) + O (z,w) where r(z,w)=−izw 2πω r −2π iω 2π iω 0 0 (z,w) = (e z, e w). 2πω r 0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 55 From Lemma 4.5 we have the representation f = ◦ f , F = O (z,w) 2πω r F for some F ∈ O (D(0,μ) ), μ> 0. We then have the following classical proposition that establishes the existence of Birkhoff Normal Forms to arbitrarily high order. Proposition 6.1. — Let ω ∈ R Q. Then, for any N ≥ 3 there exist σ -symmetric holomor- 2 2 phic maps : (C, 0) → C, Z , F : (C , 0) → C such that on a neighborhood of 0 ∈ C one N N N has (r =−izw) −1 ⎪ f ◦ ( ◦ f ) ◦ f = ◦ f Z 2πω r F F ⎨ N 0 Z N N 2(N+1) 3 (6.133) F (z,w) = O (z,w), Z (z,w) = O (z,w), N N (r) = 2πω r + O (r). N 0 Remark 6.1. — The sequences (Z ) and ( ) converge respectively in C[[z,w]] N N N N and in R[[r]].If Z ∈ C[[z,w]] and ∈ R[[r]] are their respective limits one has in ∞ ∞ C[[z,w]] the formal identity −1 f ◦ ( ◦ f ) ◦ f = Z 2πω r F ∞ 0 Z ∞ (6.134) 3 2 Z (z,w) = O (z,w), (r) = 2πω r + O (r). ∞ ∞ 0 Conversely, (6.134)deﬁnes uniquely; is the Birkhoff Normal Form BNF(f ) of f ∞ ∞ (and BNF(f ) of f ). In particular, BNF(f ) is invariant by (formal) symplectic conjugacies which are tangent to the identity. Remark 6.2. —If f = ◦ f with = (r) = 2πω r + O(r ) and F(z,w) = F 0 2(N+1) O (z,w) then N+1 (6.135)BNF(f )(r) = (r) + O (r). 6.1.2. BNF near a KAM circle (Action-Angle case). — Let f : (T × R, T ×{0}) → (T × R, T ×{0}) be a real analytic symplectic diffeomorphism of the form f (θ , r) = (θ + 2πω , r) + (O(r), O(r )).Wenoticethat : (θ , r) → (θ + 2πω , r).Wecan 0 2πω r 0 thus write f under the form (h,ρ > 0) 2h 2 f = ◦ f , F ∈ O (e (T × D(0, ρ))), F = O (r). 2πω r F σ h The normalizing map Z is unique up to composition on the left by a formal generalized symplectic rotation , ∞ A A ∈ R[[r]]. 56 RAPHAËL KRIKORIAN Proposition 6.2. — Let ω ∈ R be Diophantine. Then, for any N ≥ 3 there exist real analytic maps : (R, 0) → R, Z , F : (T × R, T×{0}) → R such that N N N −1 f ◦ ( ◦ f ) ◦ f = ◦ f Z 2πω r F F N 0 Z N N (6.136) N+1 2 2 F (θ , r) = O (r), Z (θ , r) = O (r), (r) = 2πω r + O (r). N N N 0 ω ω ω Remark 6.3. —Let C (T)[[r]] (where C (T) = C (T)) be the set of formal h>0 h power series n ω (6.137)F(θ , r) = F (θ )r , F ∈ C (T) for all n ∈ N. n n n∈N The sequence (Z ) converges in C (T)[[r]] and the sequence ( ) converges in N N N N R[[r]].If Z ∈ C (T)[[r]] and ∈ R[[r]] are their respective limits one has in ∞ ∞ C (T)[[r]] the formal identity −1 f ◦ ( ◦ f ) ◦ f = Z 2πω r F ∞ 0 Z ∞ (6.138) 2 2 Z (θ , r) = O (r), (r) = 2πω r + O (r). ∞ ∞ 0 Conversely, (6.138)deﬁnes uniquely; is the Birkhoff Normal Form BNF(f ) ∞ ∞ of f . In particular, BNF(f ) is invariant by (formal) symplectic conjugacies which are of the form id + (O(r), O(r )). Remark 6.4. —If f = ◦ f with = (r) = 2πω r + O(r ) and F(θ , r) = F 0 N+1 O (r) then N+1 (6.139)BNF(f )(r) = (r) + O (r). Remark 6.5. — The reason why we impose a Diophantine condition on ω in the statement of Proposition 6.2 is the following. The existence of the formal Birkhoff Nor- mal Form (6.136) derives from an inductive procedure where at each step n ∈ N one con- ω ω structs a formal conjugation f with Y ∈ C (T)[[r]] that conjugates f (F ∈ C (T)[[r]], Y n F n n n n ω n+1 F = O (r))to f (F ∈ C (T)[[r]],F (θ , r) = O (r)). To perform this conjugation n F n+1 n+1 n+1 step one has to solve a cohomological equation F (θ , r) = Y (θ + 2πω , r) − Y (θ , r) + n n 0 n 2π F (ϕ, r)dϕ where r is a formal variable but θ lies on T (see Lemma E.7). This equation is classically solved by passing to Fourier coefﬁcients (see for example [13]) but it involves small denominators that can be dealt with if ω satisﬁes an arithmetic condition, for example a Diophantine one (weaker conditions such as Bruno condition or even ln q = o(q ) n+1 n will also be ﬁne ). The normalizing map Z is unique up to composition on the left by a formal integrable twist of the form , ∞ A A ∈ R[[r]]. As usual p /q are the convergents of ω . n n 0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 57 6.2. Pérez-Marco’s Dichotomy. — We now discuss the extension of Pérez-Marco’s Dichotomy, Theorem 1.3, to the diffeomorphism setting. The ﬁrst part of Pérez-Marco’s argument in [36], translated in our (CC)- setting, is based on the fact that the coefﬁcients of the Birkhoff Normal Form B(r) = n n b (F)r = b (F)(−izw) of ◦ f depend polynomially on the coef- d d n n 2πω,r F n∈N n∈N k l ﬁcients of F(z,w) = F z w . More precisely, if we denote by [F] , j ≥ 3, d d k,l j (k,l)∈N ×N k l the homogeneous part of F of degree j , [F] = F z w , then the coefﬁcients j k,l |k|+|l|=j of the homogeneous part of degree 2j , [B ◦ r] = b (F)(−izw) of B ◦ r,are 2j k |k|=j polynomials of degree 2j − 2 in the coefﬁcients of [F] ,...[F] . As a consequence, 3 j if (z,w) → F(z,w), (z,w) → G(z,w) are two σ -symmetric holomorphic functions 3 3 such that F(z,w) = O (z,w),G(z,w) = O (z,w),thenfor any n ≥ 3, the maps t → b (tF+ (1− t)G) are polynomials of degree ≤ 2|n|− 2. The second argument in [36] is then to use results from potential theory (in particular the Bernstein-Walsh Lemma ) applied to the family of polynomials t → b (tF + (1 − t)G) that have a degree which behaves linearly in n. To check that the arguments of [36] adapt to the diffeomorphism case it is hence enough to check that t → b (tF + (1 − t)G) are polynomials of degree ≤ 2(|n|− 1): Lemma 6.3. — If F, G are σ -symmetric holomorphic maps F, G = O (z,w) in the (CC)- 2 d case (resp. F, G = O (r) in the (AA)-case) then, for every n ∈ N , |n|≥ 2,t → b (tF + (1 − t)G) is a polynomial of degree ≤ 2(|n|− 1) (resp. ≤|n|− 1). Proof. — We refer to Appendix E wherewediscuss formal aspectsofthe BNF (mainly in the (AA)-case) and give a proof of the lemma in Section E.3. 6.3. Approximate BNF. 6.3.1. Elliptic ﬁxed point case ((CC)-Case). — Our aim is to give a more quantitative version of Proposition 6.1. h 1/2 2 Recall that W ={(z,w) ∈ D(0, e ρ ) , −izw ∈ D(0,ρ)} and we denote h,D(0,ρ) sometimes by W the set W . h,ρ h,D(0,ρ) Let m ≥ 4 be an integer. Applying Proposition 6.1 with m = N − 1 we can assume that the diffeomorphism f is of the form f = ◦ f 0 0 (6.140) 2 2m (r) = 2πω r + O (r), and F (z,w) = O (z,w). 0 0 0 In particular (cf. Remark 2.1)for some h > 0and any ρ> 0 small enough we can assume that (6.141) F h ρ . e W h,D(0,ρ) It states that if a polynomial of degree n is bounded above by some constant M on a not pluripolar compact set m m K ⊂ C then its size at any point z ∈ C is not larger than M × exp(ng (z)) where g (z) is the Green function of K with K K pole at ∞. 58 RAPHAËL KRIKORIAN Denote by (p /q ) the sequence of best rational approximations of ω which has n n n≥1 0 the following properties (cf. [20], Chap. 5, formulae (7.3.1)–(7.3.2) and Prop. 7.4): for all n ∈ N 1 1 (6.142) <(−1) (q ω − p )< , n 0 n q + q q n n+1 n+1 and (6.143) ∀ 0 < k < q , ∀ l ∈ Z, |kω − l|≥|q α − p | > . n 0 n−1 n−1 2q We refer to Notations 2.3, 2.6 and 4.8 before stating the following proposition. Proposition 6.4. — Let a := max(2a + 1, 30) where a is the exponent that appears in Lemma 5.4 and assume that (6.141) holds for some m ≥ a . Then for any β> 0 and any n 1 1 β 1−β BNF BNF q BNF there exist g ∈ Symp (W −6 ), and functions F ∈ O (W −6 ) ∩ O (z,w), ∈ −1 −1 σ −1 ex.,σ h,q h,q n n q q q n n n −6 O (D(0, q )) such that BNF −1 BNF −6 BNF BNF (6.144) [W ] (g ) ◦ ◦ f ◦ g = ◦ f −1 F −1 h,q 0 0 F q q n n −1 −1 q q n n 1−β BNF q (6.145) (r) − BNF(f )(r) = O (r), in R[[r]] −1 BNF (6.146) 1 −1 BNF −(m−27) (6.147) g − id 1 ≤ q −1 C BNF 1−β (6.148) F ≤ exp(−q ). −1 W −6 n h,q BNF If ∈ TC(A, B) (see Notation 2.6) one can choose ∈ TC(2A, 2B). −1 Proof. — See the Appendix, Section F.2. 6.3.2. (AA) or (CC) case when ω is Diophantine. — We formulate here a more quan- titative version of the classical Birkhoff Normal Form Theorem (Propositions 6.1, 6.2) which holds both in the (AA) or (CC) cases, provided ω is Diophantine: (6.149) ∀ k ∈ Z {0}, min|kω − l|≥ (τ ≥ 1). l∈Z |k| CC AA Let as usual W be equal to either W or W and ∈ O (D(0, 1)), h,D(0,ρ) σ h,D(0,ρ) h,D(0,ρ) (r) = 2πω r + O(r ),where ω is assumed to be Diophantine with exponent τ . 0 0 We deﬁne (as before a is the constant introduced in Lemma 5.4) (6.150) a := max(2(τ + a), 12) 1,τ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 59 andweassume that forsome m ≥ a , the function F ∈ O (e W ) (h > 0) satisﬁes 1,τ σ h,D(0,1/2) 2m (CC) − Case : F(z,w) = O (z,w) (6.151) (AA) − Case : F(θ , r) = O(r ). We set (CC) − Case : b = 2(τ + 1) (6.152) (AA) − Case : b = τ + 1. Proposition 6.5. — Assume that ω satisﬁes (6.149) and that for some m ≥ a (6.151) 0 1,τ BNF b BNF holds. Then, for any β> 0 and any 0 <ρ 1, there exist ∈ O (D(0,ρ )), F ∈ β σ ρ ρ 1−β (1/ρ) BNF O (W b ) ∩ O (r) and g ∈ Symp (W b ) such that on W b one τ τ τ σ h,D(0,ρ ) h,D(0,ρ ) h,D(0,ρ ) ρ ex.,σ has BNF −1 BNF BNF BNF (6.153) (g ) ◦ ◦ f ◦ g = ◦ f ρ ρ ρ ρ 1−β BNF (1/ρ) (6.154) (r) − BNF(f )(r) = O (r), in R[[r]] BNF BNF m−10 g − id 1 ≤ ρ BNF 1−β (6.155) F exp(−(1/ρ) ). ρ b h,D(0,ρ ) BNF If ∈ TC(A, B) then ∈ TC(2A, 2B). Proof. — See the Appendix, Section F.3. Remark 6.6. — Inequality (6.155) can also be written BNF (1−β)/b F exp(−(1/ρ) ). 1/β W τ h,D(0,ρ) We note that Iooss and Lombardi (Theorem 1.4 of [24]) obtained, for a similar problem, a more precise estimate but with essentially the same exponent 1/b = 1/(1 + τ) (their 2 1/(1+τ) estimate reads ≤ (cst)ρ exp(−(cst)/ρ )). Remark 6.7. — In the (CC)-case and when ω is in DC(κ, τ ),one canprove the previous proposition (maybe not with the same value for the exponent b ) by using Propo- −1 τ sition 6.4 and the fact that q ≤ q ≤ κ q . n n+1 6.4. Consequence of the convergence of the BNF. Lemma 6.6. — Assume that BNF(f ) coincides as a formal power series with a holomorphic function ∈ O(D(0, ρ)) and, for 0 <ρ ≤ ρ,let ∈ O(D(0,ρ)) be such that N+1 (r) − BNF(f )(r) = O (r) in R[[r]] (6.156) ≤ 1. D(0,ρ) 60 RAPHAËL KRIKORIAN Then −1 − exp(−N). D(0,e ρ) ∞ ∞ N k k k Proof. — Let (z) = ξ z , (z) = b z , = ξ z and = k k N k N k=0 k=0 k=0 b z .Wehavefrom(6.156) and the fact that = BNF(f ) in R[[r]] k=0 (6.157) = . N N On the other hand, we observe that if g : z → g z is in O(D(0,ρ)) one has by k∈N k −1 Cauchy’s estimates |g |ρ ≤g ,hence for |z| < e ρ k D(0,ρ) k k g z ≤ g (z/ρ) k D(0,ρ) k≥N+1 k≥N+1 −N ≤ 2e g . D(0,ρ) As a consequence, −N −N −1 −1 − e , − e . N D(0,e ρ) D(0,ρ) N D(0,e ρ) D(0,ρ) We conclude using (6.157). To summarize, Corollary 6.7. — If BNF( ◦ f ) converges and coincide on D(0, ρ) with ∈ O(D(0, ρ)), then for any β> 0 and ρ 1 one has: –If ω is τ -Diophantine ((AA) or (CC)-case) BNF 1−β − b exp(−(1/ρ) ). D(0,ρ ) – In the (CC) case for any ω irrational 1−β BNF − −6 exp(−q ). −1 D(0,q ) n+1 n+1 n+1 7. KAM Normal Forms We present now, in the uniﬁed (AA)-(CC) framework, the KAM scheme that is central in all this paper. This will be used in Sections 10 and 11 to construct the adapted Normal Forms and in Section 12 to get estimates on the Lebesgue measure of the set of KAM circles. For the sake of clarity we break down our main result into three proposi- tions: Propositions 7.1, 7.2, 7.4. As usual we denote in the (AA)-case M = T × C,M = T× R,O = T×{0} and ∞ R in the (CC)-case M = C × C and M = M∩{r ∈ R},O={(0, 0)}. R ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 61 7.1. The KAM statement. — Let 0 < ρ< h/2 < 1/2, A ≥ 1, B ≥ 1and ∈ O (e D(0, ρ)) satisfying the following twist condition (see (2.59)): −1 2 3 (7.158) ∀ r ∈ R, A ≤ (1/2π)∂ (r) ≤ A, and (1/2π)D ≤ B. −1 h Let ω(r) = (2π) ∂(r). The image of D(0, e ρ) by ω is contained in a disk D(ω (0), 3Aρ). We can assume without loss of generality that ω(0) ∈[−1/2, 1/2] and consequently, if ρ is small enough we can assume (7.159) ω(D(0, e ρ) ∩ R)⊂[−3/4, 3/4]. Let C, a be the constants of Proposition 5.5. We introduce (7.160) a = 2(a + 2) + 10 2 0 and assume that F ∈ O (e W ) satisﬁes σ h,D(0,ρ) (7.161) F h ≤ ρ . e W h,D(0,ρ) By Cauchy’s inequality (2.53) one has j 2(a +2)+1 (7.162) ε := maxD F ≤ ρ . h,D(0,ρ) 0≤j≤3 Associated to this ε> 0 there exists a unique N > 0such that − ln ε = N/(ln N) . We then deﬁne for n ≥ 1 the following sequences that depend on ε = ε , h and ρ> 0: n−1 N = (4/3) N −hN /(ln N ) n n ε = e ⎪ n 2(a +2) −1 0 (7.163) K = ε ,(a ≥ 5) ⎪ −2 δ = 2(ln N ) h n n n−1 n−1 ρ = ρ exp(− δ ), h = h − (1/2) δ > h/2. n j n j j=1 j=1 If ρ is small enough, for all n ≥ 1 one has −1/20 −1/20 ρ ≥ e ρ, h ≥ e h n n and (cf. (7.162)), −1 (7.164) ρ /2 > 2K −1 −1 −a (7.165) (δ (2K )) K ε < C n n n (C is the constant of Proposition 5.5). 62 RAPHAËL KRIKORIAN Proposition 7.1. — Assume that and F are as above and that ρ 1. Then, with the A,B notations (7.163) the following holds: for n ≥ 1 there exist a decreasing (for the inclusion) sequence of holed domains (U ) , functions ∈ O (U ), F ∈ O (W ) with U = D(0, ρ), = , n n≥1 n σ n n σ h ,U 1 1 n n F = F and, for n ≥ 2, 1 ≤ m < n, diffeomorphisms g ∈ Symp (W ), such that: 1 m,n h ,U n n ex.,σ (7.166) satisﬁes a (2A, 2B) − twist condition, (cf. (2.59)) (7.167) g (W ) ⊂ W m,n h ,U h ,U n n m m −1 (7.168) on W , g ◦ ◦ f ◦ g = ◦ f h ,U F m,n F n n m,n m m n n 1/2 (7.169) g − id 1 ≤ ε , m,n C (7.170) maxD F ≤ ε . n W n h ,U n n 0≤j≤3 Proof. — We construct inductively for n ≥ 2 sequences U , F , , g satisfying the n n n m,n conclusion of the proposition with the additional requirements Requirement 1: For n ≥ 2, U is of the form (7.171) U = D(0,ρ ) D(c ,κ ), c ∈ R, #I ≤ 2N n n i i i n n−1 i∈I n−1 n−1 n−1 −1 −1 δ 2 1/2 δ −1 l l l=1 l=1 (7.172) K ≤ κ ≤ K e ,( κ ) ≤ 2e N K . i l n−1 1 i l i∈I l=1 Requirement 2: For n ≥ 2, ∈ O (U ) satisﬁes an (A , B )-twist condition with (see n σ n n n (2.52) for the notation a(U )) −1 −1 (7.173) 1 ≤ A ≤ 2A − K , 1 ≤ B ≤ 2B − K n n n n (7.174) 8max(ρ, a(U )) × A × B < 1. n n n n−1 1/2 1/2 (7.175) − 3 ≤ ε ≤ 2ε C (D(0,ρ)) n l 1 l=1 n−1 1/2 (7.176) and ∀ m < n, g − id 1 ≤ C ε ε (Cfrom (2.43)). m,n C l l=m For some n ≥ 1, assume the existence of U , F , and the validity of conditions n n n (7.171), (7.172), (7.173), (7.174), (7.175)(if n ≥ 2) and deﬁne ω = (1/2π) . Since (7.174) n n −1 is satisﬁed we can apply Proposition 2.7 (with A = A ,B = B ,3A ν = K , β = l/k): for n n n 2 −1 each (k, l) ∈ Z ,0 < k < N ,suchthat D(l/k,(3A K ) ) ∩ ω (U ) = ∅, there exists n n n n n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 63 (n) c ∈ R such that l/k (n) ω (c ) = l/k l/k (7.177) (n) −1 −1 ∀ r ∈ C D(c , K ), |ω (r) − (l/k)|≥ (3A K ) . n n n l/k n We denote E ={(k, l) ∈ Z , 0 < k < N , 0≤|l|≤ N , n n n −1 D(l/k,(3A K ) ) ∩ ω (U ) = ∅} n n n n and we see that (7.178)#E ≤ 2N . Note that from (7.175)and (7.159)wehave |l/k|≤ 1. Hence, if we deﬁne (n) −1 (7.179)V = U D(c , K ) n n l/k (k,l)∈E we have for any r ∈ V (cf. (7.173)) ∗ −1 ∀ (k, l) ∈ N × Z, 1 ≤ k < N %⇒ k ∂ (r) − l ≥ (6AK ) ; n n n 2π hence the non-resonance condition (5.126) (with τ = 0, K = 6AK ,N = N )issatisﬁed. n n On the other hand (7.171)–(7.172)(n ≥ 2) and (7.164)(n = 1) show using (7.179)that (recall ρ < ρ< h/2) −1 −1 (7.180)d(W ) = d(V ) = min(d(U ), K ) = K h ,V n n n n n n and (7.165)and (7.170) show that −1 −a (7.181) (δ d(W )) K F < C . n h ,V n n h ,U n n n n We can thus apply Proposition 5.5 (with τ = 0, K = 6AK , δ = δ ,N = N )on V :ifone n n n n deﬁnes (n) −δ −δ δ −1 n n n (7.182)U = e V = e U D(c , e K ) n+1 n n l/k n (k,l)∈E −δ /2 n there exist Y ∈ O(e W ),F ∈ O (W ), ∈ O (U ) such that n n+1 σ h ,U n+1 σ n+1 h ,V n+1 n+1 n n (ρ small enough) −1 (7.183) Y −δ /2 K δ F n e W n n W h ,V n h ,U n n n n −1 (7.184) W , f ◦ ◦ f ◦ f = ◦ f h ,U Y F F n+1 n+1 n n n Y n+1 n+1 n 64 RAPHAËL KRIKORIAN (7.185) = + M(F ) n+1 n n j −1 − a 2 −δ N /2 0 n n (7.186) maxD F K (δ K ) (F + e F ). n+1 W A n n n n W h ,U n W h ,U n n n+1 n+1 h ,U n n 0≤j≤3 Let us show that the Requirements 1 (7.171)–(7.172) are satisﬁed for n + 1. From (7.182)and (7.171) we see that U = D(0,ρ ) D(c ,κ ) n+1 n+1 i i i∈I n+1 −1 2 δ δ −1 n n where I ≤ I + 2N (cf. (7.178)) and for all i ∈ I ,min(e K , e K ) ≤ κ ≤ n+1 n n+1 i n n−1 n −1 2 2 δ 1/2 δ 1/2 2 −2 1/2 l n l=1 K e . Similarly, ( κ ) ≤ e (( κ ) + (2N K ) ).Inother words, 1 i∈I i i∈I i n n n+1 (7.171)–(7.172) are satisﬁed for n + 1. Let us now prove that the Requirements 2, (7.173), (7.174)(7.175) are satisﬁed for Wh n + 1 and in particular that has a nice Whitney extension := .Weﬁrst n+1 n+1 n+1 3 Wh apply Lemma 2.2 to get a C , σ -symmetric extension M(F ) : C → C for (M(F ), U ) n n n such that j Wh 3 −6 j sup D M(F ) (1 + #J ) (δ d(U )) maxD M(F ) −δ /10 . n C U n n n e U n n 0≤j≤3 0≤j≤3 In particular, using Cauchy’s inequalities, (7.171), (7.172), (7.163), (5.104)one gets j Wh 6 −1 −6 −3 (7.187) sup D M(F ) N (δ K ) δ M(F ) n C n n U n−1 n−1 n n 0≤j≤3 7 1/2 K ε ≤ ε . n−1 n From (7.185) we see that if we deﬁne the σ -symmetric function Wh (7.188) := + M(F ) n+1 n n one has n+1 n+1 n+1 −1 −1 1/3 and (7.173) , are satisﬁed (since −K + ε < −K ). To see that (7.174) holds we n+1 n+1 n n n+1 use the fact that since the second inequality in (7.172) is true for n+ 1 (as already checked) n −1/2 2 1/2 −1 one has a(U ) ≤ ( κ ) ≤ 2 N K ≤ K .If ρ is small enough we see n+1 l i l 1 i∈I l=1 n+1 that (7.162), (7.163)and (7.173) ensure the validity of (7.174) . n+1 n+1 −(3/4)δ Finally let us check (7.176) . From Lemma 2.2 we see that (Y , e W ) n+1 h ,V n n 3 Wh has a C σ -symmetric Whitney extension Y such that Wh 3 −6 j 3 −(2/3)δ (7.189) Y (1 + #J ) (δ d(U )) maxD Y n . V n n n C e V n n n 0≤j≤2 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 65 2 −1 From (7.171), (7.179), (7.172) we see that #J ≤ 2N ,d(V ) ≥ K hence using Cauchy’s V n n n n (1/2)+ −1 0+ 7 inequalities, (7.183), (7.163) and the fact that δ , N ≤ K and K ε ≤ ε ,weget n n n n n n Wh 1/2 (7.190) Y 3 ≤ ε . n n −1 If we deﬁne g = f ∈ Symp (W ) and for m ≤ n, g = g ◦ g n,n+1 Wh h ,U m,n+1 m,n n,n+1 ex,σ n+1 n+1 we have from (4.98)and (2.43) g − id 1 ≤ C(g − id 1 +g − id 1 ) ≤ C ε ε m,n+1 C m,n C n,n+1 C l m l=m which is (7.176) and implies (7.169) . n+1 n+1 −1 −1 Note that f = f on W and (7.184)shows that (7.168) and (7.167) Wh h ,U Y n+1 n+1 n+1 n+1 Y n are satisﬁed. We now check that (7.170) holds for n + 1; from (7.186) it is enough to verify a +2 2 −δ N /2 0 n n (7.191)K (ε + e ε )< ε n n+1 n n −δ N /2 n n 2(a +2) or equivalently since e = ε ,K = ε , n n n 2−(1/2) 2ε ≤ ε n+1 which is clearly satisﬁed since 3/2 > 4/3, cf. (7.163). 7.2. Localization of the holes. — We can localize the holes of the domains U : Proposition 7.2 (Localization of the holes).— For each 1 ≤ m < n, one has 1/2 (7.192) ∂ − ∂ 2 ε n m m and for some sets E ⊂{(k, l) ∈ Z , 0 < k < N , 0≤|l|≤ N } (1 ≤ i ≤ n − 1) one can write U i i i n as n−1 n−1 (i) −1 δ t=i (7.193) D(0,ρ ) D(c , s K ), s = e ∈[1, 2] n i,n−1 i,n−1 l/k i i=1 (k,l)∈E (i) (i) −1/5 where ρ ≥ e ρ and c is on the real axis and is the unique solution of the equation ω (c ) := l/k i l/k (i) −1 (2π) ∂ (c ) = l/k. l/k Proof. — Inequality (7.192) is consequence of (7.188), (7.187). The expression (7.193) comes from (7.182). We now give a more detailed description of the structure of D(U ), the set of holes of the domains U appearing in Proposition 7.2, cf. (7.193). n 66 RAPHAËL KRIKORIAN Lemma 7.3. — With the notations of Propositions 7.1–7.2: (1) For any n ≤ n , (k , l ) ∈ E ,j = 1, 2, one has 1 2 j j n (n ) (n ) 1 2 1/2 if l /k = l /k then |c − c | ε 1 1 2 2 l /k l /k n 1 1 2 2 1 (7.194) (n ) (n ) 1 2 −2 if l /k = l /k then |c − c | N . 1 1 2 2 l /k l /k n 1 1 2 2 (2) Let n , n ∈ N,n ≤ n and 0 <κ <κ be such that 1 2 1 2 2 1 −2 1/2 κ + κ N , ε κ − κ . 1 2 1 2 n n 2 1 (n ) Then, two disks D(c ,κ ), (k , l ) ∈ E ,j = 1, 2, are either disjoint or l /k = l /k and j j j n 1 1 2 2 l /k j j j (n ) (n ) 2 1 D(c ,κ ) ⊂ D(c ,κ ). 2 1 l /k l /k 2 2 1 1 Proof. — Item (1) is due to (7.192) and the fact that if l /k = l /k 1 1 2 2 −2 |(l /k ) − (l /k )|≥ 1/(k k ) ≥ N . 1 1 2 2 1 2 −2 Item (2) is a consequence of Item (1). Indeed, if l /k = l /k then since κ + κ N and 1 1 2 2 1 2 (n ) (n ) (n ) (n ) 1 2 −2 1 2 |c − c | N (we assume n ≤ n ), the disks D(c ,κ ) and D(c ,κ ) must have 1 2 1 2 l /k l /k n l /k l /k 1 1 2 2 2 1 1 2 2 an empty intersection. On the other hand, if l /k = l /k , then because of the fact that 1 1 2 2 (n ) (n ) (n ) (n ) 2 1/2 2 1/2 1 1 |c − c | ε ,the disk D(c ,κ ) contains D(c ,κ ) because ε + κ <κ . 1 2 2 1 l /k l /k n l /k l /k n 1 1 2 2 1 1 1 2 2 1 7.3. Whitney conjugation to an integrable model. — By applying Lemma 2.2 one sees −δ −δ 3 n n that (F , e W ) and (f , e W ) have C real symmetric Whitney extensions n h ,U F h ,U n n n n n Wh −δ Wh F ∈ O (e W ), f Wh ∈ Symp (W ) (the canonical map associated to F )such σ h ,U F h ,U n n n ex,σ n n n that (see the discussion leading to (7.187) and inequality (4.98)) Wh 1/2 1/3 3 Wh 1 F ε , f − id ε . C F C n n n We hence have −δ −1 (7.195)on e W , g ◦ ◦ f Wh ◦ g = ◦ f Wh . h ,U F m,n F n n m,n m m n n We show in the next Proposition that shrinking a little bit the domain of validity of the preceding formula one can impose that g leaves invariant the origin O={r = 0}∩ M . m,n R Lemma 7.4. — There exists g ∈ Symp (W −1 ) that coincides with g on m,n m,n ex,σ h /2,U D(0,K ) n n m W −1 and h /2,CD(0,K ) n m 1/4 (7.196) g ({r = 0})={r = 0}, g − id 1 ≤ ε . m,n m,n C −1 −1 Wh 3 −(1/2)δ Proof. — Recall that g = f ◦ ··· ◦ f with Y ∈ C ∩ O (e W ) m,n Wh Wh σ h ,V k k Y k n−1 satisfying (7.190). Let χ : R →[0, 1] be a smooth function with support in [−1, 1] ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 67 and equal to 1 on [−1/2, 1/2] and deﬁne the C σ -symmetric function Y = (1 − 2 Wh Wh 3 Wh 1/4 χ((K r/2) ))Y . One has Y = Y on W and Y 3 K Y 2 ≤ ε m k h,CD(0,K /2) k C C k k m m 1/4 −1 −1 −1 −1 1 hence f coincide with f on W and f − id ε . Since for m ≤ k ≤ Wh C h,CD(0,K ) k Y Y Y k k −1 n − 1, Y is null on a neighborhood of {r = 0} the diffeomorphism f ﬁxes {r = 0} and n−1 1/4 1/4 so does g . The inequality in (7.196) follows from the fact that ε ε . m;n k=m m Note that the sequence of diffeomorphisms n → g converges in C to a σ - m,n symmetric diffeomorphism g : C → C ﬁxing the origin and that satisﬁes g − m,∞ m,∞ id 1 ε . On the other hand, the sequence of diffeomorphisms (f Wh ) converges in C C m F n to the identity and from (7.192) the sequence of functions ( ) , ∈ O (U ) converges n n σ n 2 2 in C to some σ -symmetric limit ∈ C (C);hence from (7.195) ∞ σ −δ −1 on e W −1 , g ◦ ◦ f Wh ◦ g = . F m,∞ h /2,U D(0,K ) m,∞ n n m m m ∞ n≥m Recall the notations of Section 4.4 and let −δ −1 L = R ∩ e (U D(0, K )), m n n≥m −δ −1 W = M ∩ e W . L R m h /2,U D(0,K ) n n m n≥m Proposition 7.5. — For any m ≥ 1 one has −1 (7.197) on W , g ◦ ◦ f ◦ g = L F m,∞ m,∞ m m ∞ (7.198) g (W ) ⊂ W , m,∞ L R∩U 1/4 (7.199) g ({r = 0})={r = 0}, g − id 1 ≤ ε m,n m,n C 2(a +3) −2δ m 0 (7.200) Leb ((R ∩ e U ) L ) ε . R m m Proof. — Let us prove (7.198). Note that since g and g coincide on m,n m,n −δ W −1 one has from (7.167) g (e W −1 ) ⊂ W hence since g is m,n h ,U m,n h /2,CD(0,K ) h ,U D(0,K ) m m n m n n m σ -symmetric, g (W ) ⊂ W and g (L ) ⊂ W = W . m,n L R∩U m,∞ m R∩U m m m R∩U The conjugation relation (7.197) comes from the fact that Wh ◦ f Wh coincides on m m W with ◦ f . R∩U m m m For the proof of (7.200) we ﬁrst observe that from the expression (7.193), for each − δ −δ 2 −1 l n l=m n > m the set e U e U is a union of at most 2N disks of radii ≤ 2K m n n n 2 −1 hence the Lebesgue measure of its intersection with M is ≤ 4N K . In consequence, n n ∞ 2(a +3) − δ −δ 2 −1 0 l n l=m the Lebesgue measure of R ∩ e U e U is N K ≤ ε m n n≥m+1 n=m+1 n n 68 RAPHAËL KRIKORIAN hence 2(a +3) −2δ −δ 0 m n Leb (e U e U ) ε M m n n≥m −δ δ −1 −2δ n m m and since L ⊃ ( e U ) e D(0, K ) we get that Leb ((R ∩ e U ) L )) ≤ n M m m n≥m m 2(a +3) 0 δ −1 εm + e K ;(7.200) follows from this inequality. Remark 7.1. — If U is a holed domain, Propositions 7.1, 7.2, 7.5 as well as their proofs, extend without any change to the situation where F ∈ O (e W ) and σ h,U ∈ O (e U) satisﬁes the twist condition (7.158)–(2.60) and if the following smallness assumption on F holds (7.201) F h ≤ d(W ) . e W h,U h,U 8. Hamilton-Jacobi Normal Form and the Extension Property Our aim in this section is to provide a useful approximate Normal Form (that we call the Hamilton-Jacobi Normal Form) in a neighborhood of a q-resonant circle {r = c} (by which we mean that for some (p, q) ∈ Z × N , p ∧ q = 1 ω(c) = ). Let 0 < ρ< h/2 < 1/20, c ∈ R, (p, q) ∈ Z × N , p ∧ q = 1, ∈ O (D(c, 6ρ) ), F ∈ O (W ) such that σ h,D(c,6ρ) −1 −1 2 −1 3 (8.202) ∀ r ∈ R, A ≤ (2π) ∂ (r) ≤ A, and (2π) D ≤ B. a −8 (8.203) ε := F ≤ min((6ρ) ,(10A) ), h,D(c,6ρ) −1 (8.204) ω(c) := (2π) ∂(c) = 1/8 −1 (8.205) (6ρ) <(Aq) < h/10, 6ρ< |c|/4 where a is the constant appearing in Proposition G.1 of Appendix G on Resonant Nor- mal Forms. The purpose of this section is to prove the following result: Proposition 8.1 (Hamilton-Jacobi Normal Form).— Let D = D(c, ρ) . There exists a disk D, 1/33 (8.206) D := D(q c, ρ) q ⊂ D = D(c, ρ) , with ρ q≤ ε and HJ HJ HJ ∈ O (D D)), F ∈ O (W ), g ∈ Symp ((W ) σ σ q q h/9,(DD) σ h/9,(DD) D D D ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 69 such that HJ (8.207) satisﬁes a (2A, 2B) − twist condition HJ HJ −1 (8.208) W ,(g ) ◦ ◦ f ◦ g = HJ ◦ f HJ q F h/9,(DD) F D D D D HJ 1/8 (8.209) g − id 1 qε HJ 1/4 (8.210) F exp(−1/(6qρ) )ε. D h/9,(DD) Moreover, one has the following: HJ Extension property: ( , D, D) satisﬁes the following Extension Principle: If there exists a holomorphic function ∈ O(D) such that HJ − ν (4/5)D(1/5)D 1/200 then ρ q ν . Remark 8.1. — From Lemma K.1 and Remark K.1 of the Appendix, we just have to prove the Proposition in the (AA)-setting. This is the setting in which we shall work in all this section. The proof of the ﬁrst part of Proposition 8.1 is done in Section 8.7 and that of the second part (Extension Principle), based on Proposition 8.10, is done in Section 8.9. From now on we deﬁne ρ = 6ρ. 8.1. Putting the system into Resonant Normal Form. — From Proposition G.1 of the Appendix on the existence of approximate q-Resonant Normal Form, we know that there res −1/q −1/q cor −1/q exist ∈ O (D(c, e ρ)), g ∈ Symp (e W ), F , F ∈ O (e W ) σ RNF h,D(c,ρ) σ h,D(c,ρ) ex,σ res res such that F is 2π/q-periodic, M (F ) = 0, and −1 −1/q res cor e W , g ◦ ◦ f ◦ g = ◦ ◦ f ◦ f h,D(c,ρ) F RNF 2π(p/q)r F F RNF (8.211) res res F is 2π/q − periodic, M (F ) = 0, with − ( − 2π(p/q)r) −1/q F D(c,e ρ) W h,D(c,ρ) res −1/q F F e W h,D(c,ρ) h,D(c,ρ) (8.212) −1/4 cor F −1/q exp(−ρ )F e W W ⎪ h,D(c,ρ) h,D(c,ρ) −1 g − id 1 ≤ (qρ ) F RNF C h,D(c,ρ) 70 RAPHAËL KRIKORIAN Inequalities (8.212) and the fact that satisﬁes an (A, B)-twist condition on D(0, ρ) show that there exists a unique c ∈ R such that ∂ (c) = 0, |c − c| ε. 8.2. Coverings. — We denote R = R + −1[−h, h] and by j the q-covering h q j (C/(2π Z)) × C → (C/(2π/q)Z) × C (8.213) (θ + 2π Z, r) → (θ + (2π/q)Z, r). res −2/q Since the function F : (θ , r) : (R /(2π)Z) × D(c, e ρ) → C is invariant by h−2/q (θ , r) → (θ + 2π/q, r) onecan pushitdowntoafunction res res res −2/q F : (R /(2π/q)Z) × D(c, e ρ) → C, F ◦ j = F . h−2/q q j j q q Let : (C/(2π/q)Z) × C → C/(2π)Z × C (8.214) (θ , r) → (qθ, q(r − c)) res −2/q and deﬁne F : (R /(2π)Z) × D(0, e qρ) → C by qh−2 res res 2 −1 F = q F ◦ ; j q −2/q −2/q for all (θ, r) ∈ T × D(0, qe ρ) and (θ , r) ∈ T × D(c, e ρ) such that θ = qh−2 h−2/q qθ, r = q(r − c) one has res res 2 (8.215) F (θ, r) = q F (θ , c + r). Let f res be the (exact) symplectic mapping (for the symplectic form dθ ∧ d r)deﬁnedby (4.87): if ( ϕ, R) = (ϕ, R), (θ, r) = (θ , r) res ( ϕ, R) = f res (θ, r) ⇐⇒ (ϕ, R) = f (θ , r). F F If we set (8.216) (r) := q (c + (r/q)) − 2π(p/q)(r/q) 2 2 3 = (1/2)∂ (c)r + O(r ) 2 3 = r + r b(r) we have −1 res (8.217) ◦ ◦ f ◦ = ◦ f res . q F F q q ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 71 Note that since satisﬁes an (A, B)-twist condition, one has from the ﬁrst equation of (8.212) (to which one applies Cauchy’s inequality), the estimate −1/10 2 (8.218) ∀ r ∈ D(0, e ρ), ∂ (r) 1, 3 −1/10 1. C (D(0,e qρ)) 8.3. Approximation by a Hamiltonian ﬂow. — The following proposition says that up to some very good approximation ◦ fres can be seen as the time-1 map of a Hamilto- nian vector ﬁeld in the plane. vf per −2/q Proposition 8.2. — There exists F , F ∈ O (T −2/q × D(0, e qρ/2)),suchthat σ e qh/2 −2/q on T −2/q × D(0, e qρ/2) one has e qh/2 (8.219) ◦ fres = per ◦ fvf F +F F per res 1/4 res 2 (8.220) F = F + O(ρ F −2/q ) = O(q F ) T ×D(0,e qρ) h,D(0,ρ) −2/q e qh vf 1/4 −2/q −2/q (8.221) F exp(−1/(qρ) )F . h,D(0,ρ) e qh/2,D(0,e qρ/2) Proof. — This is a consequence of (8.212), (8.215) and Proposition H.1 applied to ◦ f res (since by (8.212), (8.215), condition (H.545) is satisﬁed). i 3 Let F(θ , r) = f (θ )r + r f (θ , r) and deﬁne (cf. (8.216)) i=0 per (8.222) (θ , r) = (r) + F (θ , r) 2 2 3 (8.223) =: r + f (θ ) + f (θ )r + f (θ )r + r (b(r) + f (θ , r)) 0 1 2 1 f (θ ) 1 f (θ ) 1 1 = ( + f (θ )) r + − 2 + f (θ ) 4 + f (θ ) (8.224) 2 2 + f (θ ) + r (b(r) + f (θ , r)) where −3 2 (8.225)max (|f |,|f |,|f |,|f |) (qρ) q ε. 0 1 2 −2/q T ×D(0,e qρ/2) −2/q e qh/2 8.4. From to .— We assume in the rest of this section that > 0 and we set (8.226) ρ = qρ/3. The next lemma provides a more convenient expression for the function, viewed per as a Hamiltonian, = + F which was deﬁned in (8.222). 72 RAPHAËL KRIKORIAN Lemma 8.3. — There exists a (not exact) symplectic change of coordinates G∈Symp (T × qh/3 −1/10 D(0,ρ )) of the form G(θ , r) = (θ , r − e (θ )) and ∈ O(T × D(0, e ρ )) such that q 0 qh/3 q −1 2 3 (8.227) (θ , r) := ◦ G (θ , r) = (θ)(r − e (θ ) + r f (θ , r)) with , e , e ∈ O (T ),f ∈ O (T × D(0,ρ )), 0 1 σ qh/3 σ qh/3 q −2 −1 (8.228) (·)− qρ ε, max(e ,e ), qρ ε, f 1. qh/3 0 qh/3 1 qh/3 qh/3,ρ Proof. — See Appendix L.1. Remark 8.2. — The previous lemma and (8.224) show that (θ) = + f (θ ) + O(ρ ε) and 1 f (θ ) e (θ )=− + O(ρ ε), 2 + f (θ ) 1 f (θ ) f (θ ) 1 0 e (θ )=− + + O(ρ ε). 4 ( + f (θ )) + f (θ ) 2 2 Remark 8.3. — Since is deﬁned up to an additive constant (this will not change the value of e ), we can assume that (θ , e (θ )) dθ = 0 1/2 (θ) 2π which is equivalent to the following condition that we will assume to hold from now on dθ 1/2 (8.229) (θ) e (θ ) = 0. 2π 8.5. Hamilton-Jacobi Normal Form for .— The symplectic diffeomorphism is the time-1 map of a Hamiltonian deﬁned on the cylinder, and as such, it is integrable in the Hamilton-Jacobi sense: the level lines of the Hamiltonian foliate the cylinder and naturally provide invariant curves for the Hamiltonian ﬂow. On some open sets it is possible to conjugate to a Hamiltonian depending only on the action variable: this is the Hamilton-Jacobi Normal Form; see Proposition 8.7. The purpose of this Subsection is to quantify this fact. These are cylindrical domains outside the “eyes” deﬁned by separatrices (think of a pendulum). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 73 Recall the expression for 2 3 (θ , r) = (θ)(r − e (θ ) + r f (θ , r)). Let 0 ≤ s ≤ h/3. We denote −1 (8.230) ε := e 0 qρ ε, ε = ε (s)=e , 1 1 C (T) 1,s 1 1 qsh/3 and for L 1 we introduce 1/2 1/2 (8.231) λ := Lε ,λ = λ(s, L) = Lε , 0,L s,L 1 1,s with the requirement −1/2 (8.232) λ < qρ/6 = ρ /2 or equivalently 1 L qρε . s,L q 1,s We notice that 0 <λ ρ and that from the Three Circles Theorem s,L q 1−s s (8.233) ε (0) ≤ ε (s) ≤ ε (0) ε (1) 1 1 1 1 hence 1/2 (1−s)/2 (8.234) λ = Lε ≤ λ ≤ Lε . 0,L s,L 1 1 Notation 8.4. — For 0 < a < a and z ∈ C we denote by A(z; a , a ) the annulus centered 1 2 1 2 at z with inner and outer radii of sizes respectively a and a .When z = 0 we simply denote this annulus 1 2 by A(a , a ). 1 2 Before giving the Hamilton-Jacobi Normal Form of we need two lemmas. Lemma 8.5. — There exists a holomorphic function g deﬁned on Dom(g) := (T × A(λ ,ρ )) qsh/3 s,L q 0≤s≤1 such that for every (θ , z) ∈ Dom(g) one has (8.235) (θ , g(θ , z)) = z . Moreover, there exists g ˚ ∈ O(Dom(g)) such that on Dom(g) one has −1/2 −2 ˚ ˚ (8.236) g(θ , z) = (θ) z(1 + g(θ , z)), g L . Dom(g) Proof. — See the Appendix, Section L.3. 74 RAPHAËL KRIKORIAN Since T × A(λ ,ρ ) ⊂ Dom(g) we can deﬁne the function ∈ O(A(λ ,ρ )) by 0,L q 0,L q : A(λ ,ρ ) → C 0,L q 2π −1 (8.237) (u) = (2π) g(ϕ, u)dϕ. Using (8.236) we see that can be written 2π −1 −1/2 (u) = γ u(1 + (u)), γ := (2π) (θ) dθ, −2 L . A(λ ,ρ ) s,L q Lemma 8.6. — There exists a solution H ∈ O(A(2λ ,ρ /2)) of the equation s,L q (8.238) (H(z)) = z. Moreover it can be written −1 −2 ˚ ˚ (8.239)H(z) = γ z(1 + H(z)), H ≤ L . A(2λ ,(1/2)ρ ) s,L q Proof. — See the Appendix, Section L.4. We now apply the preceding results with s = 1/2. Proposition 8.7 (Hamilton-Jacobi).— There exists an exact symplectic change of coordinates 1/32 W ∈ Symp (T × A(3ε ,ρ /3)) such that qh/7 q ex,σ 1 −1 (8.240) W ◦ ◦ W = 2 1/4 (8.241) W − id 1 qε . Proof. — Let H be the function deﬁned by the previous lemma (with s = 1/2) and deﬁne for z ∈ A(2λ ,ρ /2) and θ ∈ J := [−4π, 4π]+ i[−qh/6, qh/6] 1/2,L q qh/6 (8.242)S(θ , z) = g(ϕ, H(z))dϕ. [0,θ] We notice that by Cauchy’s Formula, (8.237)and (8.238) S(θ + 2π, z) − S(θ , z) = g(ϕ, H(z))dϕ [θ,θ+2π] 2π = g(ϕ, H(z))dϕ 0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 75 = 2π(H(z)) = 2π z hence : (θ , z) → S(θ , z) − θ z deﬁnes a holomorphic function on T × A(2λ ,ρ /2). Moreover, from (8.242), qh/6 1/2,L q (8.236)and (8.239) one can write S(θ , z) = g(ϕ, H(z))dϕ −1/2 = (ϕ) H(z)(1 + g(ϕ, H(z)))dϕ −1/2 = γθ H(z) + (ϕ) H(z)g ˚(ϕ, H(z))dϕ −1/2 = θ z(1 + H(z)) + (ϕ) H(z)g ˚(ϕ, H(z))dϕ and we see that −2 L (1 + qh/6). T ×A(2λ ,ρ /2) qh/6 1/2,L q 1/4 Deﬁne U = T × A(2Lε + δ, ρ /2 − δ) and note that by (8.234) one has L,δ qh/6−δ q 1/4 λ ≤ Lε so that U ⊂ T × A(2λ ,ρ /2) and 1/2,L L,0 qh/6 1/2,L q −2 qL . L,0 By Cauchy’s estimates −2 (8.243) 2 q(δL) . C (U ) L,δ 1/16 Let us choose, δ = ε and −7/32 (8.244)L = ε . 1/4 (−7/32)+(8/32) 1/32 (7/16)−(2/16) 5/16 −2 −2 −2 −3 We then have Lε = ε = ε ,L δ = ε = ε ,L δ = 1 1 1 1 1 (7/16)−(3/16) 1/4 ε = ε hence 1 1 1/4 1/32 (8.245) qε . C (T ×A(2ε ,ρ /2)) qh/6 q 1/32 Using Lemma 2.2 and Lemma 4.4, we see that (, T × A(2ε ,ρ /2)) has qh/6 q 2 Wh aC , σ -symmetric Whitney extension such that −1 1/32 −1 (8.246) W = f , W = f Wh ∈ Symp(T × A(3ε ,ρ /3)) Wh qh/7 q 76 RAPHAËL KRIKORIAN −4/32 1/4 and (W − id 1 ε ε ) 1 1 1/8 (8.247) W − id 1 qε . On the other hand taking the derivative of (8.242)wehave (8.248) ∂ S(θ , z) = g(θ , H(z)) and so S is a solution of the Hamilton-Jacobi equation ∂ S (8.249) (θ , (θ , z)) = (θ , g(θ , H(z))) ∂θ (8.250) = H (z)(by (8.235)). −1 Hence, the exact symplectic change of variable W = f ∂ S −1 w = = w + ∂ (θ, z) ∂θ (8.251) W = f : (θ , w) → (ϕ, z) ⇐⇒ ∂ S ϕ = = θ + ∂ (θ, z) ∂ z conjugates to since from (4.82) (θ ,w) H(z) −1 ◦ W = H ⇐⇒ W ◦ ◦ W = . This concludes the proof. 8.6. Consequences on ◦ f .— Let G : (θ , r) → (θ , r + e (θ )) be the diffeomor- F 0 phism introduced in Lemma 8.3 and (8.252) W = G ◦ W. 1/32 We notice that W ∈ Symp (T × A(3ε ,ρ /3)) and that its image contains G(T × qh/7 q qh/7 σ 1 1/32 A(3ε ,ρ /3)) (see (8.246)); from (8.247)and (8.228)wehave 1/8 (8.253) W − id qε . Corollary 8.8. — One has −1 (8.254) W ◦ ◦ f res ◦ W = 2 ◦ f vf F H F with vf 1/4 1/32 (8.255) F exp(−1/(qρ) )F . h,D(0,ρ) T ×A(4ε ,ρ /4)) qh/8 q 1 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 77 Proof. — Recall that from (8.219) and the deﬁnition of (8.222)) ◦ f res = per ◦ f vf F +F F = ◦ f vf . By Lemma 8.3 and Proposition 8.7 −1 −1 G ◦ ◦ G = , W ◦ ◦ W = 2 hence −1 W ◦ ◦ W = 2 and so −1 −1 vf 2 vf vf vf W ◦ ◦ f ◦ W = ◦ f , f = W ◦ f ◦ W F F F F which is (8.254). vf The estimate on F comes from (8.221)and (8.253). 8.7. Proof of Proposition 8.1: existence of Hamilton-Jacobi Normal Form. — Let Wbe the diffeomorphism constructed in Corollary 8.8 (cf. (8.252)). The map (deﬁned in (8.214)) 1/32 1/32 −1 sends ((R + i]− h/8, h/8[)/(2π/q)Z) × A(c; 4q ε , ρ/4) to T × A(4ε , qρ/4). qh/8 1 1 From (8.217), (8.254) one has −1 −1 res W ◦ ◦ ◦ f ◦ ◦ W = 2 ◦ f vf q H F F q hence −1 −1 −1 res ( ◦ W ◦ ) ◦ ◦ f ◦ ( ◦ W ◦ ) q q q F q −1 −1 = ( ◦ 2 ◦ ) ◦ ( ◦ f vf ◦ ). H q F q q q −1 −1 Let W, and f be lifts by j (deﬁned in (8.213)) of ◦ W ◦ , ◦ 2 ◦ ˚ HJ ˚ vf q q H q F q q −1 res res and ◦ f vf ◦ . Since ◦ f is a lift by j of ◦ f one has for some m ∈ Z q q F F F (0 ≤ m ≤ q − 1) −1 res W ◦ ◦ f ◦ W = ◦ HJ ◦ f vf ˚ ˚ 2π(m/q)r F F where HJ −2 2 vf vf ˚ ˚ (8.256) (r) = q H (q(r − c)), F = O(F ). If we deﬁne −1 cor cor (8.257) f cor = W ◦ f cor ◦ W, F = O(F ) F F HJ (8.258) g = g ◦ W (g from (8.211)) RNF RNF 78 RAPHAËL KRIKORIAN one has from (8.211) (note that W commutes with ) 2π(p/q)r HJ −1 HJ −1 per (8.259) (g ) ◦ ◦ f ◦ g = ◦ W ◦ ◦ f ◦ W ◦ f cor F 2π(p/q)r F = ◦ ◦ HJ ◦ f vf ◦ f cor ˚ ˚ 2π(p/q)r 2π(m/q)r F HJ HJ =: ◦ f with (see (8.256), (8.255), (8.212)) 1/32 HJ −1 (8.260) ∈ O (A(c; 5q ε , ρ/5)), HJ −2 2 (8.261) (r) = 2π((p + m)/q)r + q H (q(r − c)) 1/32 HJ vf cor vf cor −1 ˚ ˚ (8.262) F = F + F + O (F , F ) ∈ O (T × A(c; 5q ε , ρ/5)) 2 σ h/9 HJ 1/4 (8.263) F exp(−1/(qρ) )ε. −1 With a slight abuse of notation, we can write W = ◦ W◦ and using (8.258), (8.252) and the deﬁnition of W(cf. Proposition 8.7) we can write 1/32 HJ −1 −1 g = g ◦ ◦ (G ◦ W) ◦ ∈ Symp (T × A(c; 5q ε , ρ/5)). RNF q h/9 q σ The last inequality of (8.212)and (8.253) show that (remember (8.230)) 1/8 HJ 1− 1/8 −1 (8.264) g − id 1 qε + qε qε (ε qρ ε). C 1 HJ Note that since g − id 1 , f − id (cf. (8.203)) and f HJ − id (cf. (8.263)) are C F F 1/q, the conjugation relation (8.259) shows that the integer m appearing in (8.261) must be equal to 0. Hence, HJ −2 2 (8.265) (r) = 2π(p/q)r + q H (q(r − c)). HJ Let us now check that one can choose in O (D D) which satisﬁes a (2A, 2B)- twist condition. Indeed, from (8.239) and Cauchy’s inequality (recall our choice (8.244) −7/32 L = ε ) we see that −2 2 −2 2 q H (q(r − c)) − γ (r − c) 1/32 3 −1 C (T ×A(c;6q ε ,ρ/6)) h/9 −3/32 7/16 11/32 qε ε ≤ qε . 1 1 1 −6/32 11/32 5/32 We now apply Lemma 2.2: since ε × ε ε , there exists a C σ -sym- 1 1 1 1/7 1/8 3 −2 2 metric Whitney extension with C -norm less that qε < ε for (q H (q(r − c)) − 1/32 −2 2 −1 1/8 γ (r − c) , T × A(c; 6q ε , ρ/6)). Using (8.265) and the inequality ε h/9 HJ (1/2) min(A, B) (cf. (8.203)) we see that has a Whitney extension (that we still de- HJ note )suchthat HJ (8.266) satisﬁes a (2A, 2B) − twist condition. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 79 We can now conclude the proof of Proposition 8.1. We deﬁne the disk Dof Propo- sition 8.1 as (cf. (8.230), (8.228)) 1/33 D = D(q c, ρ) q ⊂ D(c, ε ) (8.267) 1/32 1/32 1/33 −1 q c = c, ρ q= 6ε = 6e ≤ ε (ε qρ ε) 1 0 1 C (T) and the disk D can be taken to be (recall |c − c| ε) (8.268) D = D(c, ρ/6) = D(c, ρ) . With these notations, the set of conclusions (8.207)–(8.210) are consequences of (8.266), (8.259), (8.264)and (8.263). 8.8. Extending the linearizing map inside the hole. — In general the previously deﬁned maps g,, H are not holomorphically deﬁned on a whole disk but rather on an annulus with 1/32 inner disk of radius 6ε where ε =e 0 . In this subsection we quantify to which 1 1 C (T) extent the domains of holomorphy of these maps can be extended if one knows that the HJ frequency map coincides on this annulus with a holomorphic function deﬁned on a disk (containing the annulus). Notation 8.9. — In the following we denote by C(0, t) the circle of center 0 and radius t > 0. Proposition 8.10. — If there exists a holomorphic function deﬁned on D(0,ρ ) such that (8.269) − H ≤ ν C(0,ρ /2) then (1/6)− ε =e 0 ν . 1 1 C (T) We prove this proposition in Section 8.8.2. We now take s = 0.(cf. (8.231)) By (8.236)for z ∈ A(λ/2,λ/4), |g(θ , z)| compares to λ andthusfrom(8.235)and (8.227) 2 2 3 z = (θ) g(θ , z) − e (θ ) + O(g ) 1 80 RAPHAËL KRIKORIAN so that 1/2 2 3 (8.270) g(θ , z) = z / (θ ) + e (·) + O(g ) 1/2 2 2 (8.271) = (z / (θ )) + e (θ ) + O(λ ). Let’s introduce 1/2 (8.272) g(θ , z) = + e (θ ) (θ) 2π −1 −1 (8.273) (·) = (2π) g(θ ,·)dθ, H = where the inverse is with respect to composition. The functions and Hare deﬁned on 1/2 {z ∈ C, Lε < |z|} for some ﬁxed L 1, independent of ε , satisfying −1/2 (8.274)L ≤ (ρ /2)ε (wetakehere s = 0, cf. (8.231)). 8.8.1. Computation of a residue. 1/2 Lemma 8.11. — For any circle C(0, t) centered at 0 with Lε < t <ρ /2 one has 1 dθ 2 3/2 2 zH(z) dz = (γ /4) (θ) e (θ ) 2π i 2π C(0,t) T 2π −1 −1/2 where γ = (2π) (θ) dθ . Proof. — We compute the expansion of g(θ ,·) (cf. (8.272)) into Laurent series: on 1/2 C D(0, Lε ): 1/2 1/2 −2 g(θ , z) = (z/ (θ ) ) 1 + (θ)e (θ )z 1 1 1/2 −2 2 −4 −6 = (z/ (θ ) ) 1+ (θ)e (θ )z − ( (θ )e (θ )) z +O(z ) 1 1 2 8 z 1 1 1/2 −1 3/2 2 −3 −5 = + (θ) e (θ )z − (θ) e (θ ) z + O(z ). 1 1 1/2 (θ) 2 8 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 81 2π −1 As a consequence since (z) = (2π) g(θ , z)dθ we have with the notation γ = 2π −1 −1/2 (2π) (θ) dθ the identity −1 −3 −5 (z) = γ(z + a z + a z ) + O(z ) −1 −3 where 2π −1 −1 1/2 a = γ (1/2)(2π) (θ) e (θ )dθ −1 1 (8.275) 2π −1 −1 3/2 2 a = γ (−1/8)(2π) (θ) e (θ ) dθ. −3 1 By our choice (8.229)wehave a = 0 and we can thus write −1 (8.276) = ◦ (id + u) where z = γ z and −3 −5 u(z) = a z + O(z ). −3 If v is deﬁned by (id + u) ◦ (id + v) = id we have −3 −4 v(z)=−a z + O(z ) −3 and therefore 2 −3 −4 2 (8.277) (z + v(z)) = (z − a z + O(z )) −3 2 −2 −3 = z − 2a z + O(z ). −3 Now since H is the inverse for the composition of (cf. (8.273)), z = ( ◦ H)(z), −1 −1 −1 we have by (8.276) H = (id + u) ◦ = (id + v) ◦ and we get by (8.277) γ γ 2 −2 2 2 −2 −3 H(z) = γ z − 2a γ z + O(z ) −3 and thus 2 −2 3 2 −1 −2 zH(z) = γ z − 2a γ z + O(z ). −3 82 RAPHAËL KRIKORIAN 1/2 HencebyCauchy’sformula and(8.275), for any circle C(0, t),Lε < t <ρ /2: 2 2 zH(z) dz=−2a γ −3 2π i C(0,t) dθ 3/2 2 = (γ /4) (θ) e (θ ) . 2π 8.8.2. Proof of Proposition 8.10. 1/2 Lemma 8.12. — Let Lε ≤ λ<ρ /2, L 1 (independent of ε ). One has for z ∈ q 1 A(λ/4,λ/2) 2 2 3 (8.278) |H(z) − H(z) | λ . Proof. — For z ∈ A(λ/4,λ/2), θ ∈ T one has by (8.271), (8.272) |g(θ , z) − g(θ , z)| λ so (cf. (8.237), (8.273)) (8.279) |(z) − (z)| λ . On the other hand, from Lemma L.1 g(θ , z) − g(θ , z ) 2 2 −3/L 2/L e ≤ ≤ e z − z hence (z) − (z ) 2 2 −3/L 2/L (8.280) e ≤ ≤ e . z − z Since z = (H(z)) = (H(z)) and H(z), H(z) z (cf. (8.239)), one has from (8.279) |(H(z)) − (H(z))| λ and so from (8.280) |H(z) − H(z)| λ . Since from (8.239) |H(z) + H(z)| λ we thus have 2 2 3 |H(z) − H(z) | λ . ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 83 We recall that ε =e 0 . The function − H satisﬁes (cf. (8.269), (8.239)) 1 1 C (T) 2 2 − H ν, − H 1/2 1. C(0,ρ /2) C(0,Lε ) Let M > 5and 1/2 1/2 1/M 1−1/M 1−1/M (8.281) λ := (ρ /2) (Lε ) ≤ (Lε ) M q 1 1 (we can assume ρ ≤ 1). By the Three Circles Theorem, 2 1/M − H ν . C(0,λ ) Lemma 8.12 tells us that 2 1/M 3 − H ν + λ C(0,λ ) M M hence for any z in the circle C(0,λ ) 2 1/M 3 |z(z) − zH (z)| λ (ν + λ ) and 2 2 1/M 3 (z(z) − zH (z))dz λ (ν + λ ). M M 2π i C(0,λ ) Since z → z(z) is holomorphic on D(0, 2λ ), z(z)dz = 0 and by Lemma 8.11 C(0,λ ) we get 3/2 2 2 1/M 3 (θ )e (θ ) dθ λ (ν + λ ). M M Since (θ) 1 this gives 2 2 1/M 3 e (θ ) dθ λ (ν + λ ) M M hence remembering (8.281) 5/2 1/(2M) e 2 λ ν + λ 1 L (T) M (1/2)(1−1/M) (5/4)(1−1/M) (1−1/M) 1/(2M) (5/2)(1−1/M) L ε ν + L ε . 1 1 If we deﬁne (5/4)(1−1/M)−1 (1/2)(1−1/M) (5/2)(1−1/M) (1−1/M) 1/(2M) δ = L ε ,μ = L ε ν M M 1 1 −1 this can be written (recall that e 0 = ε qρ ε)for some C > 0 1 C (T) 1 e 2 ≤ Cδ e 0 + Cμ 1 L (T) M 1 C (T) M 84 RAPHAËL KRIKORIAN and we are in position to apply Lemma M.2 (our choice M > 5 implies that for some 2β β> 0, δ ≤ ε 1): −1 2 −1 ε =e ≤ (μ /δ ) + Ch exp(−h/(Cδ ))qρ ε 1 1 C (T) M M (μ /δ ) + exp(−(1/ε ))(β> 0) M M 1 (μ /δ ) + (1/2)ε M M 1 which gives 1−((3/4)(1−1/M)) −(3/2)(1−1/M) 1/(2M) ε L ε ν or equivalently (3/4)(1−1/M) −(3/2)(1−1/M) 1/(2M) ε L ν and taking M = 5+, one ﬁnally gets: −(2−) (1/6)− ε L ν (1/6)− ≤ ν . This completes the proof of Proposition 8.10. 8.9. Proof of Proposition 8.1: the Extension Property. — From (8.265) we see that if there exists a holomorphic function deﬁned on Dsuch that HJ − ν (4/5)D(1/5)D there exists a holomorphic function deﬁned on D(0,ρ ) (recall that ρ = qρ/3 cf. q q (8.226)) such that − H ν C(0,ρ /2) and thus by Proposition 8.10 (1/6)− ε =e 0 ν . 1 1 C (T) 1/200 Now (8.267) shows that the conclusion of Proposition 8.1 holds with D = D(c,ν ). 9. Comparison Principle for Normal Forms In this section, if 0 ≤ ρ <ρ ,wedenoteby A(c; ρ ,ρ ) the annulus {z ∈ C,ρ ≤ 1 2 1 2 1 |z − c| <ρ } (it is thus the disk D(c,ρ ) if ρ = 0). 2 2 1 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 85 Proposition 9.1 ((AA) Case).— There exist positive constants C, a , a for which the following 4 5 holds. Let 0 <ρ <ρ (resp. ρ = 0 <ρ ), ε, ν > 0 and for j = 1, 2, ∈ O (A(c; ρ ,ρ )), 1 2 1 2 j σ 1 2 F ∈ O (W ),g ∈ Symp (W ) such that: , satisfy an (A, B)-twist condi- j σ h,A(c;ρ ,ρ ) j h,A(c;ρ ,ρ ) 1 2 1 2 σ 1 2 tion (2.59)and −1 g − id ≤ ε< C h j C (9.282) F ≤ ν j W h,A(c;ρ ,ρ ) 1 2 and on g (A(c; ρ ,ρ )) ∩ g (A(c; ρ ,ρ )) one has 1 1 2 2 1 2 −1 −1 g ◦ ◦ f ◦ g = g ◦ ◦ f ◦ g . 1 F 2 F 1 1 1 2 2 2 Then, if δ> 0 satisﬁes − a (9.283) Cε ≤ δ/4 <(ρ − ρ ) and Cδ ν< 1, 2 1 there exists γ ∈ R, |γ|≤ Cε such that one has − a ∂ (·+ γ) − ∂ ≤ Cδ ν. 1 2 A(c;ρ +δ,ρ −δ) 1 2 (9.284) −a (resp. ∂ (·+ γ) − ∂ ≤ Cδ ν.) 1 2 D(c,ρ −δ) Furthermore, if g and g are exact symplectic on M , one can choose γ = 0. 1 2 R Proof. — We only treat the case ρ > 0(the case ρ = 0 is done similarly). 1 1 From (9.282) we see that there exists C > 0 such that one has on W := T × 1 h−Cε A(c; ρ + Cε, ρ − Cε) 1 2 (9.285) g ◦ = ◦ g ◦ f 1 2 where −1 g := g ◦ g ∈ Symp (W ) 1 1 2 σ −1 −1 F ∈ O(W ), f := g ◦ f ◦ g ◦ f , F ν. 1 F F W 2 F 1 We write (9.286) g(θ , r) = (θ + u(θ , r), r + v(θ , r)) −1 and we introduce the notations ω = ∂ , i = 1, 2 (we drop the usual factor (2π) ). We i i have g ◦ (θ , r) = (θ + ω (r) + u(θ + ω (r), r), r + v(θ + ω (r), r)) 1 1 1 1 86 RAPHAËL KRIKORIAN and ◦ g = (θ + u(θ , r) + ω (r + v(θ , r)), r + v(θ , r)) We thus have on W := T × A(c; ρ + Cε + δ, ρ − Cε − δ) 2 h−Cε−Bρ −δ 1 2 ω (r + v(θ , r)) − ω (r) = I + u(θ + ω (r), r) − u(θ , r) 2 1 1 (9.287) v(θ + ω (r), r) − v(θ , r) = II −b with max(I ,II ) = O(δ ν). We observe that from the twist assumption on W W 1 2 2 there exists a set R ⊂ A(c; ρ + Cε + δ, ρ − Cε − δ) of Lebesgue measure δ , which 1 2 is a countable union of disks centered on the real axis, such that one has for any r ∈ A(c; ρ + Cε + δ, ρ − Cε − δ) Rand any k ∈ Z 1 2 l δ (9.288)min|ω (r) − 2π |≥ l∈Z k k so that the second identity in (9.287)gives forany r ∈ A(c; ρ + Cε + δ, ρ − Cε − δ) R 1 2 the following inequality on T (where h = h − Cε − Bρ ) h −2δ 1 2 −3 −b (9.289) v(·, r) − v(θ , r)dθ δ δ ν. h −2δ We now notice that there exists 0 ≤ t ≤ δ such that R∩ ∂ A(c; ρ + Cε + δ + t,ρ − Cε − 1 2 δ − t) =∅. The maximum principle applied, for any ϕ ∈ T , to the holomorphic h −2δ function v(ϕ,·) − v(θ ,·)dθ deﬁned on A(c; ρ + Cε + δ + t,ρ − Cε − δ − t) shows 1 2 that (9.289) holds for any r ∈ A := A(c; ρ + Cε + 2δ, ρ − Cε − 2δ).Wethushave 2δ 1 2 −(4+b) (9.290) ∂ v = O(δ ν). θ h −3δ,A 1 3δ Taking the ∂ derivative of the ﬁrst line of (9.287) and using the previous inequality show that (from now on the value of b may change from line to line) −b ∂ u(θ + ω (r), r) − ∂ u(θ , r) = O(δ ν). θ 1 θ By the same argument used to establish (9.290)weget −b (9.291) ∂ u = O(δ ν) θ h −4δ,A 1 4δ (wehaveused thefactthat ∂ u(θ , r)dθ = 0). Since g is symplectic on W ,det Dg(θ , r) ≡ θ 1 1hence (1 + ∂ u(θ , r))(1 + ∂ v(θ , r)) − ∂ u(θ , r)∂ v(θ , r) = 1 θ r r θ andinviewof(9.290), (9.291) −b ∂ v = O(δ ν) r h −4δ,A 1 4δ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 87 which combined with (9.290) implies, −b (9.292) v − γ = O(δ ν), γ = v(0, 0) ∈ R. h −4δ,A 1 4δ The ﬁrst equation of (9.287) implies that −b ω (·+ γ) − ω (·) = O(δ ν) 2 1 A 4δ which is the ﬁrst conclusion of the Proposition (9.284). If g and g are exact symplectic, g is also exact symplectic and one can write g = f 1 2 Z for some Z = O (g − id) which means g(θ , r) = (ϕ, R) if and only if r = R + ∂ Z(θ , R), 1 θ ϕ = θ + ∂ Z(θ , R). In particular r = r + v(θ , r) + ∂ Z(θ , r + v(θ , r)) and since Z(θ , r + v(θ , r)) = ∂ Z(θ , r + v(θ , r)) + ∂ Z(θ , r + v(θ , r)∂ v(θ , r) θ R θ dθ we get from (9.290) −b v(θ , r)=− Z(θ , r + v(θ , r)) + O(δ ν) dθ which after integration in θ yields −b v(θ , r)dθ = O(δ ν). We can now conclude from (9.292)that −b γ = O(δ ν). In particular, taking γ = 0 does not affect the estimate (9.284). Proposition 9.2 ((CC)-Case).— Under the assumptions of the previous Proposition 9.1: (1) If c = 0, ρ = ρ , ρ = 0 and g , g are exact symplectic then 2 1 1 2 −a ∂ (·) − ∂ (·) ≤ Cδ ν. 1 2 D(0,ρ −δ) (2) If ρ < |c|/4 then all the conclusions of the previous Proposition 9.1 are valid. Proof. — The proof of Item (2) follows from Item (2) of Lemma K.1 of the Ap- pendix applied to Proposition 9.1. 88 RAPHAËL KRIKORIAN So we concentrate on the proof of Item (1), c = 0, ρ = ρ , ρ = 0. We use the 2 1 −1 symplectic change of coordinates of Section K (θ , r) = ψ (z,w), ± −2h 2h ψ : T × (0,ρ) → W ± ∩{e < |z|/|w| < e } ± h h, (0,ρ) α α −1 −1 where α< π/10. Setting g = ψ ◦ g ◦ ψ , g = g ◦ g , g (θ , r) = (θ + u (θ , r), r + j,± j ± ± 1,± ± ± ± 2,± v (θ , r)) we are then reduced to the preceding situation where g is replaced by g ,the ± ± annulus A(c ; ρ ,ρ ) is replaced by the angular sector (ρ − 4δ) and h by h − 4δ, 2 1 2 α+4δ so that (9.287) holds on T × (ρ − 4δ). Like in the previous case, one can ﬁnd h−4δ α−4δ 0 ≤ t ≤ δ such that the Diophantine condition (9.288) holds for any r ∈ (0; t,ρ − α+4δ 4δ − t) := (ρ − 4δ) ∩ A(0; t,ρ − 4δ − t). Still by the Maximum Principle (9.290) α+4δ holds on T × (0; t,ρ − 5δ − t) with v replaced by v and one can conclude as h−5δ ± α+5δ we’ve done before that (9.291) holds with u replaced by u as well. Finally this gives the existence of γ = v (0, 0) ∈ R such that on (0; t,ρ − 5δ − t) ± ± α+5δ −b (9.293) ω (·+ γ ) − ω (·) = O(δ ν). 2 ± 1 A 4δ Now, if g and g areexact symplectic thesameistrue for g , g (cf. Remark 4.2)and 1 2 1,± 2,± hence g is also exact symplectic; we can thus prove, like in the proof of Proposition 9.1, −b that ±γ = O(δ ν). We can hence assume that γ = 0 in equation (9.293). Since α< + − π/10 we deduce that on A(0; t,ρ − 5δ − t) = (0; t,ρ − 5δ − t) ∪ (0; t,ρ − α+5δ α+5δ 5δ − t) one has −b ω (·) − ω (·) = O(δ ν). 2 1 A(0;t,ρ−5δ−t) But ω ,ω ∈ O(D(0,ρ)), hence by the Maximum Principle 1 2 −b ω (·) − ω (·) = O(δ ν). 2 1 D(0,ρ−5δ−t) 10. Adapted Normal Forms: ω Diophantine Recall that for τ ≥ 1, κ> 0 l κ DC(κ, τ )={ω ∈ R, ∀ k ∈ Z , min|ω − |≥ } 0 0 1+τ l∈Z k |k| DC(τ ) = DC(κ, τ ). κ>0 10h 10h Let h > 0, 0 < ρ< 1, A, B ≥ 1, ∈ O (e D(0, ρ)),F ∈ O (e W ) such σ σ h,D(0,ρ) that −1 −1 2 −1 3 (10.294) ∀ r ∈ R, A ≤ (2π) ∂ (r) ≤ A, and (2π) D ≤ B −1 (10.295) ω := (2π) ∂(0) ∈ DC(κ, τ ) ⊂ DC(τ ) 0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 89 (10.296) ∀ 0 <ρ ≤ ρ, F 10h ≤ ρ , e W h,D(0,ρ) (10.297) where m = max(a , a + 4, a , a , 2b + 10) 1,τ 2 3 4 τ (a , a , a , a are the constants appearing in Propositions 6.5, 7.1, 8.1, 9.1 and b is 1,τ 2 3 4 τ deﬁned by (6.152)). We as usual denote ω = (1/2π)∂ (ω(0) = ω ). 10.1. Adapted KAM domains. — We use in this section the notations of Section 7, in particular we denote j a (10.298) ε := maxD F ≤ ρ . h,D 0≤j≤3 Assumption (10.296) allows us to apply Proposition 7.1 on the existence of a KAM Nor- mal Form on the domain W . We can thus deﬁne holed domains U and maps F , 2h,D(0,ρ) n n , g satisfying the conclusions of Proposition 7.1. n m,n (ρ) 10.1.1. Deﬁnition of the domains U .— Let 0 <β 1and μ ∈]1, 1 + 1/τ[ such that (10.299) μ = 1 + (1 − β) ∈]1, 1 + 1/τ[. deﬁn. We deﬁne for ρ< ρ/4 two indices i (ρ), i (ρ) ∈ N as follows: − + (10.300) i (ρ) = max{i ≥ 1, D(0, 2ρ) ∩ U = D(0, 2ρ)} − i deﬁn. and i (ρ) is the unique index such that (10.301) μ μ μ 2 (N ) ≤ N <(4/3) (N ) ≤ N . i (ρ) i (ρ) i (ρ) − + − i (ρ) We also deﬁne ι(ρ) ∈ R by ι(ρ) ∈ R such that (10.302) −ι(ρ) −1/ι(ρ) ρ = (N ) , N = ρ . i (ρ) i (ρ) − − The next lemma shows how N and N compare with ρ . i (ρ) i (ρ) − + Lemma 10.1. — One has −1 −1 (10.303) (1 + 1/τ ) + O(| ln ρ| ) ≤ ι(ρ) ≤ (1 + τ) + O(| ln ρ| ). In particular, −μ/ι(ρ) (10.304)N ρ , i (ρ) + 90 RAPHAËL KRIKORIAN where for ρ 1 1 μ (10.305) − 2β ≤ ≤ 1 − (β/2). τ ι(ρ) Proof. — To prove (10.303)wejusthavetocheck that −(1+τ) −(1+1/τ ) (10.306) (N ) ρ (N ) . i (ρ) i (ρ) − − See the details in Appendix I.1. We shall say that the domains U , i (ρ) ≤ i ≤ i (ρ),are ρ -adapted KAM domains. i − + For t > 0and i (ρ) ≤ i ≤ i (ρ) we deﬁne − + (t) (t) U = U ∩ D(0, t), D (U ) = D(U )={D ∈ D(U ), D∩ D(0, t) = ∅} i t i i i i U being the domains of Proposition 7.1 and where as usual D(U) denotes the holes of the holed domain U (see Section 2.3.1). By (7.193) (j) (t) i−1 −1 U := U ∩ D(0, t) = D(0, t) D(c , s K ), i j,i−1 i l/k j j=1 (k,l)∈E (10.307) i−1 m=j s = e ∈[1, 2] j,i−1 where (j) E ⊂{(k, l) ∈ Z , 0 < k < N , 0≤|l|≤ N },ω (c ) = l/k. j j j j l/k One can in fact in formula (10.307) restrict the union indexed by j to the set j ∈[i (ρ), i− 1]∩ N; cf. Lemma I.1 of Appendix I. (t) One can also describe U by means of its holes: (t) (10.308)U := U ∩ D(0, t) = D(0, t) D D∈D (U ) t i this decomposition being minimal. In particular, if D, D ∈ D (U ) the inclusions D ⊂ D , t i D ⊂ D do not occur. Proposition 10.2. — Let i (ρ) ≤ i < i ≤ i (ρ). − + (1) The holes D ∈ D (U ) are pairwise disjoint. (3/2)ρ i (2) If D ∈ D (U ), D ∈ D (U ) one has either D ∩ D =∅ or D ⊂ D. (3/2)ρ i (3/2)ρ i (3) The number of holes of U intersecting D(0,ρ) satisﬁes (10.309)#{D ∈ D(U ), D ∩ D(0,ρ) = ∅} ρN . (4) Let D ∈ D (U ) and deﬁne ρ i (ρ) i =−1 + min{i : i (ρ) < i ≤ i (ρ), ∃D ∈ D (U ), D ⊂ D}. D − + ρ i (ρ) + ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 91 −1 Then, D is of the form D = D(c , s K ), s ∈[1, 2], c ∈ R,ω (c )∈{l/k,(k, l) ∈ D D D D i D i D E } and one has D ⊂ U . i i D D (5) Let b be deﬁned by (6.152)(where τ is such that (10.295) is satisﬁed). One has (10.310) D(0,ρ ) ⊂ U . i (ρ) Proof. — We refer to Appendix I.2 for the proofs of Items 1, 2 and 4. Proof of Item 3 on the number of holes. — From (10.307)wejusthavetocheck that for N ∈ N 2 2 #{(k, l) ∈ Z , l/k ∈]ω − s,ω + s[, 0 < k < N, 0≤|l|≤ N} sN . 0 0 If (k, l) belongs to the preceding set one has |l − kω | < sNand thus (k, l) belongs to 2 2 2 [−N, N] ∩{(x, y) ∈ R , |x − ω y|≤ sN} a set which has Lebesgue measure sN .We 2 2 2 2 thus have for large N, #(Z ∩[−N, N] ∩{(x, y) ∈ R , |x − ω y|≤ sN} sN . Proof of Item 5, inclusion (10.310). — Recall that b ≥ τ + 1. Since ω is in DC(τ ), τ 0 −(1+τ) for (k, l) ∈ E , j ≤ i (ρ) − 1 one has |l/k − ω | N . Since satisﬁes a (2A, 2B)- j + 0 j (j) −(1+τ) −(1+τ) −1 −1 twist condition (2A) ≤ ∂ω ≤ 2A one has |c | N and because K N j l/k j j j (j) −(1+τ) −(1+τ) −1 −1 (cf. (7.163)) one has |c |− 2K N ≥ C N ,for some C > 0. Now (7.193) l/k j j i (ρ) −(τ+1) −1 shows that U contains a disk D(0, C N ) and we observe that from (10.305), i (ρ) i (ρ) (τ + 1)(μ/ι(ρ)) < τ + 1 ≤ b hence b −1 (τ+1)μ/ι(ρ) (10.311) D(0,ρ ) ⊂ D(0, C ρ ) ⊂ U . i (ρ) 10.1.2. Covering the holes with bigger disks. — Let us deﬁne (compare with (7.163)) ln N N /(ln N ) i i (10.312) K = N K e i i and for any D ∈ D := D (U ) set ρ ρ i (ρ) −1 D = D(c , K ), D ={D, D ∈ D }. D ρ ρ Notice that for any a > 0, ρ 1and i (ρ) ≤ i ≤ i (ρ) one has a − D + 1/a −1 (10.313) ε K |c |/4. D D Indeed, the inequality of the RHS is due to the fact that |c | >ρ (cf. Proposition 10.2, −1 1/(1+τ) Item 5) combined with the fact that N ρ (cf. (10.306)). i (ρ) The inequality of the LHS is a consequence of (7.163). Let us mention that these disks D are the ones on which we shall later perform a Hamilton-Jacobi Normal Form as described in Proposition 8.1. 92 RAPHAËL KRIKORIAN Lemma 10.3. — The elements of D are pairwise disjoint and for any D ∈ D one has ρ ρ D ⊂ (1/10)D ⊂ 6D ⊂ U , D (1/10)D ⊂ U . i i (ρ) D + Proof. — Let D and D be two distinct elements of D .ByProposition 10.2,Item 1, −2 −1 −1 −2 D∩ D =∅ hence from Lemma 7.3,Item 1 |c − c | N . Since K + K N D D i (ρ) i i i (ρ) + D + −1 −1 we get that D(c , K ) ∩ D(c , K )=∅. D D i i Let us now prove 6D ⊂ U .If 6D is not a subset of U one has for some i i D D −1 −1 −2 D ∈ D(U ), (6D) ∩ D = ∅ hence |c − c |≤ 6K + K N . We can apply i D D D i i i (ρ) D + 1/2 Lemma 7.3,Item 1 to deduce |c − c | ε ; but this implies that D ∩ D = ∅,hence D D i (ρ) D = D (we can apply Proposition 10.2,Item 1, since D, D ∈ D ) and by Proposi- (3/2)ρ tion 10.2,Item 4 we obtain D ⊂ U : a contradiction. Let us prove the second inclusion D (1/10)D ⊂ U . If this is not the case then i (ρ) −1 −2 for some D ∈ D(U ) one has D ∩ (D (1/10)D) = ∅ hence |c − c | K N i (ρ) D D i i (ρ) D + which implies as before using Lemma 7.3 that D = D . But since D ⊂ (1/10)D this leads to a contradiction (otherwise D ∩ (D (1/10)D)=∅). Remark 10.1. — Let us mention (this will be useful in the proof of Theorem 12.3) that 1/2 |D ∩ R| ≤ 1. D∈D 10.1.3. No-Screening Property. — Our key proposition is the following. Proposition 10.4. — For any D ∈ D(U ) such that D ∩ D(0,ρ) = ∅ the triple i (ρ) b −1 −1 (U , D (1/10)D, D(0,ρ /2)) is (10b ) | ln ρ| -good (in the sense of Deﬁnition 3.3). i (ρ) τ Proof. — From Remark 3.1 it is enough to prove that for some U ⊂ U con- i (ρ) b b τ τ taining both D(0,ρ ) and D (1/10)D, the triple (U , D (1/10)D, D(0,ρ /2)) is −1 −1 (10b ) | ln ρ| -good. Lemma 10.5. — There exists a constant C > 0 such that for any 1 ≤ s ≤ 4/3, there exists ρ ∈[sρ, sρ + 10Cρ ] such that D(0,ρ ) ∩ U = D(0,ρ ) D. i (ρ) D∈D(U ) i (ρ) D⊂D(0,ρ ) −1 −2 Proof. — From Lemma 7.3 the holes of D(U ) are C N -separated (some i (ρ) + 1 i (ρ) −1 2μ/ι(ρ) C > 0), hence for some C > 0they are C ρ -separated (cf. (10.304)) and because 1 2 −1 2 of (10.305)theyare C ρ -separated for some C > 0. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 93 −1 However, each of these disks has a radius ≤ 2K ρ . Since they are centered i (ρ) on the real line the conclusion follows. From the previous lemma we deduce the existence of a ρ ∈[(5/4)ρ , (4/3)ρ] such that all the holes D ∈ D(U ) of U intersecting D(0,ρ ) are indeed included in i (ρ) i (ρ) + + D(0,ρ ). We then set U = U ∩ D(0,ρ ) = D(0,ρ ) D. i (ρ) D∈D(U ) i (ρ) D⊂D(0,ρ ) b −1 From (10.310)wehave D(0,ρ ) ⊂ U and for any D = D(c , K ) ∈ D(U ) such D i (ρ) i + that D ∩ D(0,ρ) = ∅ one has D ⊂ D(0,(5/6)ρ ): indeed, since D ∩ D(0,ρ) = ∅, −1 −1 4 4 |c | <ρ + K <ρ + ρ hence |c |+ K <ρ + 2ρ <(5/6)ρ . On the other D D i i D D hand, from Lemma 10.3 D (1/10)D ⊂ U (D ⊂ D(0,ρ )). In this situation we can −1 −1 apply Corollary 3.4 with U = U ,B = D(0,ρ /2), d = K , ε = 2K : the triple i i i i D D (U , D (1/10)D, D(0,ρ /2)) is A-good with ln(6/5) (10.314)A = − (I) b | ln ρ| where i (ρ)−1 −1 ln(K /(20ρ )) (I) := #C (ρ) −1 ln(2K /(ρ )) i=i (ρ) with C (ρ) = #{D ∈ D(U ), D ∩ D(0,ρ) = ∅, i = i}. i i (ρ) D From (10.309)ofProposition 10.2,(10.302), (10.303), (10.312), (7.163) one has i (ρ)−1 ι(ρ) + −1 ln(K N /30) i (ρ) 2 − (I) ≤ ρ N ι(ρ) −1 ln(2K N ) i i (ρ) i=i (ρ) − i (ρ)−1 ι(ρ) −(ln N ) + ln(N /30) i (ρ) ≤ ρ N ι(ρ) −(1/(2(a + 2)))hN /(ln N ) + ln(N /2) 0 i i i (ρ) i=i (ρ) − i (ρ)−1 1+β/2 ρ (N ) (ρ 1) i β i=i (ρ) and since N is exponentially growing with i, 1+β/2 (I) ρ × (N ) . i (ρ) + 94 RAPHAËL KRIKORIAN D(0,ρ) −1 D = D(c , K ) D(0,ρ /2) R-axis −1 D = D(c , sK ) FIG. 9. — Adapted KAM Normal Forms (ω Diophantine) in the complex r -plane. The triple (ρ) b −1 (U , D (1/10)D, D(0,ρ /2)) is C | ln ρ| -good From (10.304)and (10.305)wethusget 1−(1+β/2)μ/ι(ρ) β /4 (10.315) (I) ρ ≤ ρ and from (10.314), if ρ 1 1 1 ≤A(someC > 0). 10b | ln ρ| 10.2. Coexistence of KAM, BNF and HJ Normal Forms on the adapted KAM domain. Notation 10.6. — If W is a σ -symmetric holed domain, we denote by NF (W ) (resp. h,U σ h,U NF (W )) the set of triples (, F, g) with ∈ O (U), F ∈ O (W ),g ∈ Symp (W ) ex,σ h,U σ σ h,U h,U (resp. g ∈ Symp (W )). h,U ex,σ Proposition 10.7 (Adapted Normal Forms).— Let ∈ O (U) and F ∈ O (W ) satisfy σ σ h,U (10.294), (10.295), (10.296). For any β 1 deﬁne i (ρ), μ and i (ρ) according to (10.300), τ − + (10.299)and (10.301). Then for any ρ 1 the following holds: (KAM): Adapted KAM Normal Form (Proposition 7.1). Let D ∈ D (U ). ρ i (ρ) −1 (10.316) [W ] g ◦ ◦ f ◦ g = ◦ f h,U F 1,i (ρ) F i (ρ) 1,i (ρ) ± i (ρ) i (ρ) ± ± ± ± −1 (10.317) [W ] g ◦ ◦ f ◦ g = ◦ f h,U F i ,i (ρ) F i (ρ) i ,i (ρ) i i D + i (ρ) i (ρ) + D + D D + + −1 (10.318) [W ] g ◦ ◦ f ◦ g = ◦ f . h,U F i (ρ),i F i i (ρ),i i (ρ) i (ρ) − D i i D − D − − D D 1/2 m/2 (10.319) g − id 1 ε ≤ ρ 1,i (ρ) C + ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 95 1/2 (10.320) g − id 1 ≤ ε i ,i (ρ) C D + i (1/τ )−2β (10.321) F exp(−(1/ρ) ). i (ρ) W + (ρ) h,U Note that ( , F , g ) ∈ NF (W ) and ∈ TC(2A, 2B). i i i ex,σ h,U i (HJ): Hamilton-Jacobi Normal Form. (Proposition 8.1). For any D ∈ D (U ) there exists D ⊂ ρ i (ρ) HJ HJ HJ D and ( , F , g ) ∈ NF (W ) such that σ q h/9,DD D D D HJ HJ −1 HJ HJ (10.322) (g ) ◦ ◦ f ◦ g = ◦ f [W ] F q i i F h/9,DD D D D D D D HJ 1/9 (10.323) g − id 1 ε HJ (10.324) ∈ TC(2A, 2A) HJ (10.325) F exp(−(1/ρ)). D h/9,(DD) HJ The triple ( , D, D) satisﬁes the Extension Principle of Proposition 8.1. (BNF): Birkhoff Normal Form (Proposition 6.5): BNF BNF BNF There exists ( , F , g ) ∈ NF (W bτ ) such that ex,σ h,D(0,ρ ) ρ ρ ρ BNF −1 BNF BNF BNF (10.326) (g ) ◦ ◦ f ◦ g = ◦ f ,(W bτ ) F F h,D(0,ρ ) ρ ρ ρ ρ BNF m−10 (10.327) g − id ρ . BNF (10.328) ∈ TC(2A, 2B) BNF 1−β (10.329) F exp(−(1/ρ) ) ρ τ h,D(0,ρ ) Proof. — KAM: This is just the content of Proposition 7.1. For inequality (10.321) we note that from (7.170), (7.163), (10.304), (10.305) F exp(−N /(ln(N )) ) i (ρ) h,U i (ρ) i (ρ) + i (ρ) + + −(μ/ι(ρ)) exp(−ρ ) (1/τ )−2β exp(−(1/ρ) ). −1 HJ: Let D ∈ D (U ) where D = D(c , s K ), ω (c ) = p/q, q ≤ N , p ∧ q = 1, be ρ i (ρ) D D i D i + i D D one of the disks obtained in Proposition 10.2,Item 4. By Lemma 10.3 the disk 6D = −1 −1 D(c , 6K ) is included in U . We observe that 6K < |c |/4(cf. (10.313)). Since D i D i D i D D −1 −1 −8 −1 a min(6K ,|c |/4) = 6K <(Aq) and F ε <(6K ) D i h,6D i i i D D i D D D 96 RAPHAËL KRIKORIAN (the last inequality comes also from (10.313)) condition (8.205), (8.203) are satisﬁed and we can apply Proposition 8.1 on Hamilton-Jacobi Normal Forms to ◦ f on the i i D D −1 domain W ⊂ W with ρ = K : there exists a disk D ⊂ D h,D h,U i i −1 (10.330) D := D(c ,ρ ) ⊂ (1/10)D := D(c ,(1/10)K ) ⊂ U q q D i D D i HJ HJ HJ and ( , F , g ) ∈ NF (W ) satisfying (10.322) σ q h/9,DD D D D HJ 1/8 1/9 (10.331) g − id 1 qε ≤ ε i i D D D HJ 1/4 (10.332) F exp(−(K /N ) ). W i i q D D D h/9,(DD) ln N To obtain inequality (10.325) we observe that since K = N with i (ρ) ≥ i ≥ i (ρ) i + D − D i we get −1+ln N i (ρ) −(K /N ) −N . i i D D i (ρ) Because for ρ small enough −1 + ln N ≥ 4(2 + τ) we get i (ρ) 1/4 2+τ −(K /N ) −N i i D D i (ρ) which yields, using (10.302)and (10.303)(ρ 1) 1/4 (2+τ)/ι(ρ) −(K /N ) < −(1/ρ) i i D D < −(1/ρ). BNF: We observe that D(0,ρ ) ⊂ D(0,ρ) and apply Proposition 6.5 to (, F) on e W (we use the smallness condition (10.296)). h,D(0,ρ) 10.3. Comparision Principle. — We now use the result of Section 9 to show that these various Normal Forms match to some very good order of approximation. Lemma 10.8 (Comparing Adapted Normal Forms).— For any β 1,and ρ 1 τ β BNF (1/τ )−3β (10.333) − b ≤ exp(−(1/ρ) ) i (ρ) (1/2)D(0,ρ ) + ρ −2 and for any D ∈ D there exists γ ≤ K ρ D HJ (1/τ )−3β (10.334) − (·+ γ ) ≤ exp(−(1/ρ) ). i (ρ) D (4/5)D(1/5)D Proof. — 1) Proof of (10.333). From (10.326), (10.316) and the fact that W b ⊂ W b ∩ W τ τ h,D(0,ρ ) h,D(0,ρ ) h,U i (ρ) + ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 97 BNF one has on g (W b ) ∩ g (W b ) τ τ 1,i (ρ) h,D(0,ρ ) h,D(0,ρ ) + ρ −1 BNF BNF −1 g ◦ ◦ f ◦ (g ) = g ◦ BNF ◦ f BNF ◦ (g ) . 1,i (ρ) F 1,i (ρ) F + i (ρ) i (ρ) + ρ ρ + + ρ ρ b b τ τ We can then apply Propositions 9.1–9.2 with ρ = ρ , ρ = 0, δ = ρ /2, ε = 2 1 min(m/2,m−10) (1/τ )−2β ρ , ν = exp(−(1/ρ) ) because from (10.329), (10.327), (10.319), (10.321) one sees that condition (9.283)reads min(m/2,m−10) b b b −a (1/τ )−2β τ τ τ 4 Cρ ≤ ρ /4 <ρ and C(ρ /2) exp−(1/ρ) )< 1 BNF and is satisﬁed for ρ 1(cf. (10.297)). Since g and g are exact symplec- 1,i (ρ) BNF −(b +1)a (1/τ )−2β τ 5 tic we then get − τ ≤ Cρ exp(−(1/ρ) ) which is i (ρ) D(0,(1/2)ρ ) (1/τ )−3β ≤ exp(−(1/ρ) ) if ρ is small enough. 2) Proof of (10.334). Similarly, from (10.317), (10.322) one has on the set HJ g (W ) ∩ g (W ) i ,i (ρ) h/9,D(1/5)D D + h/9,D(1/5)D HJ HJ −1 −1 HJ HJ g ◦ ◦ f ◦ (g ) = g ◦ ◦ f ◦ (g ) i ,i (ρ) F i ,i (ρ) D + i (ρ) i (ρ) D + D F D + + D D andfrom(10.320)(10.321), (10.323), (10.325), we see that Propositions 9.1–9.2 apply 1/9 −1 −1 −1 with c = c , ε = ε , δ = K /20, ρ = (1/10)K , ρ = K < |c |/4 since condition i 1 2 i D i i i i D D D D D (9.283) is implied by 1/9 −1 −1 a (1/τ )−2β Cε ≤ K /80 < K /10 and C(20K ) exp(−(1/ρ) )< 1 i i i D D D which is satisﬁed (cf. (7.163), (10.312)) if ρ is small enough. We then get for some γ ∈ R, 1/9 −2 |γ | Cε ≤ K (cf. (10.313)) that on the annulus (4/5)D (1/5)D one has | − D i (ρ) i i + D D HJ (1/τ )−3β (·+ γ )|≤ exp(−(1/ρ) ). 11. Adapted Normal Forms: ω Liouvillian (CC case) 10h 10h Let h > 0, 0 < ρ< 1, A, B ≥ 1, ∈ O (e D(0, ρ)),F ∈ O (e W ) such σ σ h,D(0,ρ) that −1 −1 2 −1 3 (11.335) ∀ r ∈ R, A ≤ (2π) ∂ (r) ≤ A, and (2π) D ≤ B −1 (11.336) ω := (2π) ∂(0) ∈ R Q 10h (11.337) ∀ 0 <ρ ≤ ρ, F ≤ ρ , e W h,D(0,ρ) where (11.338) m = 4 + max(a , 2000Aa , a , a ) 1 2 3 4 (a , a , a , a are the constants appearing in Propositions 6.4, 7.1, 8.1, 9.1). 1 2 3 4 98 RAPHAËL KRIKORIAN Using the notations of Section 6.3.1,let (p /q ) be the sequence of convergents n n n of ω : 1 1 (11.339) ≤|ω − (p /q )|≤ . 0 n n 2q q q q n+1 n n+1 n (11.340) ∀ 0 < k < q , ∀ l ∈ Z, |ω − (l/k)| > . n 0 2kq We assume that n is large enough and we set 10A (11.341) ρ = ≤ ρ/10. q q n+1 n We introduce j m−3 2000Aa (11.342) ε := maxD F (10ρ ) ≤ ρ . W n 2h,D(0,10ρ ) n 0≤j≤3 11.1. Adapted KAM domains. — Since Condition (7.161) is satisﬁed we can apply Proposition 7.1 (with ρ = 10ρ ) and deﬁne holed domains U , functions ,F , ω n i i i Prop. 7.1 i etc. In particular for 0 < t i−1 (j) −1 U ∩ D(0, t) = D(0, t) D(c , s K ), i j,i−1 j=1 (k,l)∈E l/k j (11.343) i−1 m=j s = e ∈[1, 2] j,i−1 where (j) E ⊂{(k, l) ∈ Z , 0 < k < N , 0≤|l|≤ N },ω (c ) = l/k. j j j j l/k Note that from (11.342) and the deﬁnition (7.163)of K −1 1000A 2(a +2) (11.344)K ≤ ε ≤ ρ . j n Lemma 11.1. — Let j be such that N < q /(10A) and (k, l) ∈ E . j n+1 j (1) If (k, l) ∈ Z(q , p ) one has l/k = p /q and n n n n −1 (2A) (2A) (j) 2 −1 (11.345) (40A ) ρ ≤ ≤|c |≤ ≤ ρ /5. n n p /q n n 2q q q q n+1 n n+1 n (2) If (k, l)/∈ Z(q , p ) n n (j) (11.346) |c |≥ 4ρ . l/k Proof. — Item 1 comes from (11.339) and the twist condition (7.166). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 99 To prove Item 2 we observe that if (k, l)/∈ Z(q , p ) n n l l p p 1 1 99A n n |ω − |≥| − |−|ω − |≥ − ≥ ≥ 9Aρ 0 0 n k k q q kq q q q q n n n n n+1 n n+1 (j) and from the twist condition (7.166)weget |c |≥ 4ρ . l/k ∗ − For n ∈ N deﬁne i as the unique index i such that (11.347)N − ≤ q < N − i −1 i n n and i as the unique index (see the deﬁnition of the sequence N in (7.163)) such that (3/4)q q n+1 n+1 (11.348) ≤ N + < . 2 2 (10A) (10A) We deﬁne − − (i ) (i ) −1 n n − + c = c , D := D(c , s K ), D = D(c ,|c |/24) n n n n n p /q p /q i ,i −1 n n n n n n (n) (11.349)U := U + ∩ D(0,ρ ). Note that from (11.345) 2 −1 (11.350) (40A ) ρ ≤|c |≤ ρ /5. n n n Proposition 11.2. — For n large enough, (1) D(0,ρ ) ⊂ U . (n) (2) One has D := D(U )={D }. ρ n (3) One has the following inclusion 6D ⊂ U . −6 (4) One has D(0, q ) ⊂ U + . n+1 i −6 (n) (5) The triple (U , D (1/10)D , D(0, q /2)) is 1/(10| ln ρ |)-good (in the sense of Deﬁ- n n n n+1 nition 3.3). Proof of Item 1.— If j < i and (k, l) ∈ E one has 0 < k < N ≤ q hence from j n i −1 n n (j) (j) −1 (11.346) |c |≥ 4ρ and from (11.344) |c |− 2K ≥ 3ρ . The conclusion then follows n n l/k l/k j from (11.343) applied with i = i . Proof of Item 2.— From Item 1, equality (11.343) can be written i −1 (j) −1 U ∩ D(0,ρ ) = D(0,ρ ) D(c , s K ). n n j,i−1 n l/k j (k,l)∈E j=i j n 100 RAPHAËL KRIKORIAN (i ) We observe that (q , p ) ∈ E and from (11.345), (11.344) one sees that D = D(c , n n i n n p /q n n −1 − + s − + K ) ⊂ D(0,ρ ). More generally, if (k, l) ∈ E , i ≤ j ≤ i − 1and (k, l)/∈ − n j i ,i −1 n n n n (j) 2 −1 Z(q , p ), one has N ≤ q /(10A) and (11.346), (11.344) give that D(0, c , 2K ) ∩ n n j n+1 l/k j (j) −1 − + D(0,ρ )=∅. Since the sets D(0, c , s K ), i ≤ j ≤ i , form a nested decreasing n q ,i −1 p /q n j n n + n n (for the inclusion) sequence of disks one gets U + ∩ D(0,ρ ) = D(0,ρ ) D . n n n Proof of Item 3.— This comes from the fact that |c |+ 6|c |/4 ≤ ρ . n n n −6 Proof of Item 4.— This comes from Item 1 and the fact that |c |−|c |/4 ≥ q as n n n+1 is clear from the LHS inequality of (11.345). −1 Proof of Item 5.— Notice that from (11.345)5 ≤ ρ /|c |≤ 40A and that 2K ≤ n n − 1000A ρ . We use Corollary 3.4; we have to evaluate ln(|c |/(4ρ )) ln(|c |/8ρ ) n n n n I = − −6 −1 ln(q /(2ρ )) ln(2K /ρ ) n − n n+1 ln(20) ln(320A ) ≥ − 7| ln ρ | (1000A − 1)| ln ρ | n n ≥ . 10| ln ρ | 11.2. Adapted Normal Forms. Proposition 11.3. — Let ∈ O (U) and F ∈ O (W ) satisfy (10.294), (10.295), σ σ h,U (10.296). Let 0 <β 1 and n 1 such that (11.351) q ≥ q . n+1 (KAM): Adapted KAM Normal Form ((Proposition 7.1)): One has ( , F , g ∈ NF (W ), i i i ex,σ h,U ∈ TC(2A, 2B) and −1 (11.352) g ◦ ◦ f ◦ g = ◦ f [W ] ± F F h,U 1,i ± ± ± 1,i i i i n n n −1 − + (11.353) g ◦ ◦ f ◦ g = ◦ f [W (n)] − + F F − − i ,i + + h,U n n i ,i n n i i i i n n n n 1/2 m/3 + − + (11.354) g − id 1 ,g − id 1 ε ≤ ρ C C 1,i i ,i n n n n 1−β (11.355) F exp(−q ). i (n) n+1 h,U (HJ): Hamilton-Jacobi Normal Form (Proposition 8.1). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 101 D(0,ρ ) ρ 1/(q q ) n n+1 n D = D(c , c /4) n n n −6 D(0, q ) n+1 R-axis 0 c ρ n n a/3 D = D(c ,ρ ) n n n n −6 FIG. 10. — Adapted KAM Normal Forms (CC Case) in the complex r -plane. The triple (U , U , D(0, q )) is n+1 1/(10| ln ρ |)-good HJ HJ HJ There exists ( , F , g ) ∈ NF (W ) such that n n n h/9,D D n n HJ −1 HJ HJ HJ (11.356) (g ) ◦ ◦ f ◦ g = ◦ f [W ] F q − − n n F h/9,(D D ) n n n n i i n n 1/9 HJ m/9 (11.357) g − id 1 ε ≤ ρ n n HJ (11.358) ∈ TC(2A, 2A) (1/4)−β HJ (11.359) F exp(−q ). n q n+1 h/9,(D D ) n n HJ The triple ( , D, D) satisﬁes the Extension Principle of Proposition 8.1. (BNF): Birkhoff Normal (Proposition 6.4): BNF BNF BNF −6 There exists ( , F , g ) ∈ NF (W ) such that −1 −1 −1 ex,σ h,D(0,q ) q q q n+1 n+1 n+1 n+1 BNF −1 BNF (11.360) (g ) ◦ ◦ f ◦ g = BNF ◦ f BNF [W −6 ] −1 −1 F F h,D(0,q ) q q −1 −1 n+1 n+1 n+1 q q n+1 n+1 −(m−27) BNF (11.361) g − id q −1 −6 n+1 h,D(0,q ) n+1 n+1 BNF (11.362) ∈ TC(2A, 2B) −1 n+1 1−β BNF (11.363) F ≤ exp(−q ). −1 −6 n+1 h,D(0,q ) n+1 n+1 102 RAPHAËL KRIKORIAN Proof. — KAM: This is Proposition 7.1. Inequality (11.355) comes from the corre- + + + + sponding (7.170) ε ≤ exp(−N /(ln N ) ) and the fact that N q . i i i i n+1 n n n n HJ: By Proposition 11.2,Item 3,the disk 6D = D(c ,|c |/4) is included in U . Since n n n 1/8 −1 a − − (6(|c |/24)) <(Aq ) , and F ε <(|c |/4) n n i h,6D i n n n n (the ﬁrst inequality is a consequence of (11.351) and the second of (11.342)and thefact that |c | ρ )(8.205), (8.203) are satisﬁed and we can apply Proposition 8.1 on Hamilton- n n Jacobi Normal Forms to ◦ f on the domain W ⊂ W with ρ =|c |/24: there F h,U n − − h,D − i i i n n n exists a disk D ⊂ D n n (11.364) D := D(c ,ρ ) ⊂ (1/10)D ⊂ D = D(c ,|c |/24) ⊂ U − n q q n n n n D D n n n HJ HJ HJ and ( , F , g ) ∈ NF (W ) such that one has (11.356)and σ q n n n h/9,D D n n HJ 1/8 1/9 (11.365) g − id q ε ≤ ε W n − − D h/9,(D D ) i i n n n n n HJ 1/4 (11.366) F exp(−1/(q |c |/24) ) W n n D h/9,(D D ) n n n −1 and since |c | (q q ) (n 1) n n n+1 β HJ (1/4)−β F exp(−q ). n+1 D h/40,(D D ) n n n 1/10 BNF: Since F ≤ ρ (cf. (11.337)) we can apply Proposition 6.4 on the exis- e W h,D(0,ρ ) n tence of approximate BNF in the CC case (with n + 1 in place of n): for 0 <β 1and BNF BNF BNF n 1: there exists ( , F , g ) ∈ NF (W −6 ) such that β −1 −1 −1 ex,σ h,D(0,q ) q q q n+1 n+1 n+1 n+1 BNF −1 BNF [W −6 ] (g ) ◦ ◦ f ◦ g = BNF ◦ f BNF −1 F −1 h,D(0,q ) F q q n+1 −1 −1 n+1 n+1 q q n+1 n+1 with 1−β BNF F ≤ exp(−q ). −1 W −6 n+1 h,q n+1 n+1 11.3. Comparision Principle. — These various Normal Forms match to some very good order of approximation. Lemma 11.4 (Comparing Adapted Normal Forms).— One has for any β 1,n 1 1−β BNF (11.367) + − −6 exp(−q ) −1 i (1/2)D(0,q ) n+1 n+1 n+1 −m m/2 and there exists γ q |c | such that n n n+1 HJ (1/4)−β (11.368) + − (·+ γ ) exp(−q ) i (4/5)D (1/5)D n+1 n n n n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 103 Proof. — Let us prove estimate (11.367). From (11.360)–(11.352), we see that on BNF g (W −6 ) ∩ g (W −6 ) one has −1 1,i h,D(0,q h,D(0,q n+1 n+1 n+1 BNF BNF −1 −1 BNF BNF + + g ◦ ◦ f ◦ (g ) = g ◦ ◦ f ◦ (g ) . −1 −1 F F 1,i + + 1,i n n q q −1 −1 i i n n n+1 q q n+1 n+1 n+1 1−β −6 −7 We then apply Proposition 9.2 with c = 0, ρ = 0, ρ = q , δ = q , ν = exp(−q ) (cf. 1 2 n+1 n+1 n+1 − min(m/3,m−27) (11.363), (11.355)), ε = q (cf. (11.361), (11.354)) (estimates (9.282)and (9.283) n+1 − min(m/3,m−27) 1/2 −7 −6 7 a are satisﬁed since Cq ≤ q /4 q and q exp(−q ) 1). n+1 n+1 n+1 n+1 n+1 Estimate (11.368) is a consequence of Proposition 9.2 applied to (11.353)and (11.356) −1 HJ HJ −1 g − + ◦ ◦ f ◦ (g − + ) = g ◦ HJ ◦ f HJ ◦ (g ) i ,i + + i ,i n n n n n F n n n i i n n with A(c; ρ ,ρ ) = D (1/10)D , c = c , ρ =|c |/40, ρ =|c |/4, δ =|c |/10, 1 2 n n n 1 n 2 n n n 1/5 m/9 ν = exp(−q ) (cf. (11.359), (11.355)), ε = ρ (cf. (11.354), (11.357)). Estimates n+1 m/9 (9.282)and (9.283) are satisﬁed since Cρ ≤|c |/40 |c |/5(cf. (11.350)) and n n (1/4)− −a −1 C(|c |/10) exp(−q )< 1(recall that |c | (q q ) ). n n n n+1 n+1 12. Estimates on the measure of the set of KAM circles We refer to Section 4.4 for the notations of this section. We observe (W ) := W ∩ M = W := {r ∈ U ∩ R}∩ M . h,U R h,U R U∩R R In particular in the (AA)-case (W ) = W = T × (U ∩ R) and in the (CC*)-case h,U R U∩R 2 2 2 (W ) = W ={(x, y) ∈ R ,(1/2)(x + y ) ∈ U ∩ R }. h,U R U∩R + 12.1. Classical KAM estimates. — We ﬁrst state a variant of the classical KAM the- orem on abundance of invariant circles which is a consequence of Propositions 7.1, 7.2, 7.5 and Remark 7.1 on KAM Normal Forms. In the next theorem the constant a is the one of Proposition 5.5 and the constant a was deﬁned in Section 7 by (7.160). Theorem 12.1. — Let U be a holed domain with disjoint holes D ∈ D(U) such that 1/2 (12.369) |D ∩ R| ≤ 1 D∈D(U) and ∈ O (U) ∩ TC(A, B) (cf. (7.158)) with A, B satisfying (2.60), F ∈ O (W ) σ σ h,U ε := F ≤ d(U) . h,U 104 RAPHAËL KRIKORIAN Then, if f = ◦ f one has 1/(2(a +3)) Leb (W −1/10 L(f , W )) (F ) . M e U∩R U∩R W R h,U Proof. — See Appendix J.2. Notation 12.2. — We deﬁne for ρ> 0, D (0,ρ) = D(0,ρ) ∩ R =] − ρ, ρ[ and 1/2 m (ρ) = Leb (W L(f , D (0, e ρ))). f M D (0,ρ) R R R 12.2. Estimates on the measure of the set of invariant circles: ω Diophantine (AA) or (CC) Case. — We use the notation of Section 10 and assume that (both in the (AA) or (CC)- cases) (10.294), (10.295)(10.296) hold. We denote (12.370) D = D(U ). ρ i (ρ) Theorem 12.3. — For any β> 0, ρ 1 (1/τ )−β (AA)-case m (ρ) exp(−(1/ρ) ) + |D ∩ R|. ◦f D∈D (1/τ )−β (CC) or (CC*)-case m (ρ) exp(−(1/ρ) ) + |D ∩ R |. ◦f + D∈D Moreover, for any D ∈ D one has −β 1+τ (12.371) |D ∩ R| exp(−(1/ρ) ). Proof. — If S ⊂ C we denote S = S ∩ R (if c ∈ R, D (c, t) = D(c, t) ∩ R =]c − R R t, c + t[). 1/4 1/3 Choose (cf. Lemma 10.5) ρ ∈[e ρ, e ρ] (ρ 1) such that (ρ ) U := D(0,ρ ) ∩ U = D(0,ρ ) D i (ρ) D∈D(U ) i (ρ) D⊂D(0,ρ ) hence (ρ ) −1/10 −1/10 (12.372) e D (0,ρ ) ⊂ e U ∪ (1/4)D . R R D∈D HJ Let us denote for short f = ◦ f , f = HJ ◦ f HJ and i F i i D D D HJ L = L(f , W ), L = L(f , W ). (ρ ) q i (ρ) i (ρ) D + + D D D U R R R ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 105 We have from (10.318)(10.322)(10.316)ofProposition 10.7 −1 (12.373) W , g ◦ f ◦ g = f h,U i (ρ) i (ρ),i i i i (ρ),i − − D D D − D HJ HJ HJ −1 (12.374) W ,(g ) ◦ f ◦ g = f h/9,DD D D D D −1 (12.375) W , g ◦ f ◦ g = f h,U i (ρ) i (ρ),i (ρ) i (ρ) i (ρ) i (ρ),i (ρ) − − + + + − + −1 (12.376) W , g ◦ f ◦ g = f . h,U 1,i (ρ) i (ρ) − − i (ρ) 1,i (ρ) − − From Lemma 10.3,Remark 10.1 and estimate (10.321) on the one hand, and estimate HJ (10.325) on the other hand, we see that we can apply Theorem 12.1 to f and f to i (ρ) + D get the following decompositions (12.377)W L ⊂ B , W −1/10 L ⊂ B ∪ E (ρ ) q q −1/10 i (ρ) i (ρ) e D D + + R DD D e U with B ⊂ W ,E = W and −1/10 q q q q DD D D D e D R R R (1/τ )−β/2 (12.378)max Leb (B ), Leb (B ) exp(−(1/ρ) ) M i (ρ) M R + R DD (12.379)Leb (E ) Leb (W ). −1/10 M q M q R D R e D We now introduce HJ (12.380) L := g (L ), L := g ◦ g (L ) i (ρ) i (ρ),i (ρ) i (ρ) i (ρ),i + − + + D − D D D HJ (12.381) B = g (B ), B = g ◦ g (B ), i (ρ) i (ρ),i (ρ) i (ρ) q i (ρ),i q + − + + DD − D D DD HJ (12.382) E = g ◦ g (E ). q q D i (ρ),i D − D Lemma 12.4. — One has −1/10 g (W ) L ⊂ B e D (0,ρ ) i (ρ),i (ρ) R − + with L = L ∪ L B = B ∪ (B ∪ E ). q q i (ρ) D i (ρ) + + DD D D∈D D∈D ρ ρ Proof. — We observe that from (12.372) one has −1/10 (12.383)W ⊂ W (ρ ) ∪ W e D (0,ρ ) −1/10 (1/4)D R R e U D∈D ρ 106 RAPHAËL KRIKORIAN hence g (W −1/10 ) e D (0,ρ ) i (ρ),i (ρ) R − + ⊂ g (W ) ∪ g (W ). (ρ ) −1/10 (1/4)D i (ρ),i (ρ) i (ρ),i (ρ) R − + e U − + D∈D HJ 1/9 Note that by Proposition 10.7 one has max(g − id 1 ,g − id 1 ) ≤ ε i (ρ),i (ρ) C C − + i (ρ) D − −1 2 K (since N ≤ N )hence i D i (ρ) D − HJ g (W ) ⊂ W ⊂ g ◦ g (W −1/10 ) (1/4)D (1/2)D e D i (ρ),i (ρ) R R i (ρ),i R − + − D D which yields g (W −1/10 ) e D (0,ρ ) i (ρ),i (ρ) R − + HJ ⊂ g (W ) ∪ g ◦ g (W ) ∪ E . (ρ ) −1/10 q q −1/10 i (ρ),i (ρ) i (ρ),i e D D D − + e U − D D R R D∈D We then conclude using (12.377) and the notations (12.380). 1/10 Lemma 12.5. — For some G ⊂ W one has L = L(f , G) and e D (0,ρ ) i (ρ) R − (1/τ )−β (12.384)Leb (B) exp(−(1/ρ) ) + Leb (W ). −1/10 M M q R R e D D∈D Proof. — We observe that from (4.86) (12.385) L := g (L ) = L(f , g (W )) (ρ ) i (ρ) i (ρ),i (ρ) i (ρ) i (ρ) i (ρ),i (ρ) + − + + − − + HJ HJ (12.386) L := g ◦ g (L ) = L(f , g ◦ g (W )) D i (ρ),i D i (ρ) i (ρ),i q − D D − − D D D D R R hence, L = L(f , G) i (ρ) with HJ G = g (W ) ∪ g ◦ g (W ) (ρ ) i (ρ),i (ρ) i (ρ),i q − + − D D D D U R R D∈D and clearly G ⊂ W 1/10 . e D (0,ρ ) To get the estimate on the measure of Bwe use (12.378)and (12.381)toget (1/τ )−β/2 Leb (B )) exp(−(1/ρ) ), M i (ρ R + ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 107 and (remember (10.309), (10.304), (10.305)) 2 (1/τ )−β/2 Leb B N exp(−(1/ρ) )) M q R DD i (ρ) D∈D (1/τ )−β exp(−(1/ρ) ); moreover (see (12.379), (12.382)) Leb E Leb (W ). q −1/10 q M M R D R e D D∈D D∈D ρ ρ Summing up these estimates yields the desired inequality on the measure of B. End of the proof of Theorem 12.3. Lemmata 12.4 and 12.5 give 1/10 g (W ) L(f , W ) ⊂ B D (0,ρ ) i (ρ) e D (0,ρ ) i (ρ),i (ρ) R − R − + hence 1/10 (12.387) g ◦ g (W ) g (L(f , W )) ⊂ g (B). D (0,ρ ) i (ρ) e D (0,ρ ) 1,i (ρ) i (ρ),i (ρ) R 1,i (ρ) R 1,i (ρ) − − + − − −1 Since the conjugation relation g ◦ f ◦ g = f holds on W (cf. (12.376)) 1,i (ρ) i (ρ) U ∩R − − 1,i (ρ) i (ρ) − − 1/10 and since W ⊂ W (recall that by deﬁnition (10.300) D(0, 2ρ) ⊂ W e D (0,ρ ) U ∩R h,U R i (ρ) i (ρ) − − and that g − id ρ ) one has by (4.86) 1,i (ρ) L(f , g (W 1/10 )) = g L(f , W 1/10 ). e D (0,ρ ) i (ρ) e D (0,ρ ) 1,i (ρ) R 1,i (ρ) − R − − Equation (12.387) then implies that 1/10 g ◦ g (W ) L(f , g (W )) ⊂ g (B). D (0,ρ ) e D (0,ρ ) 1,i (ρ) i (ρ),i (ρ) R 1,i (ρ) R 1,i (ρ) − − + − − Finally, inclusions W ⊂ g ◦ g (W ) and g (W 1/10 ) ⊂ D (0,ρ) D (0,ρ ) e D (0,ρ ) R 1,i (ρ) i (ρ),i (ρ) R 1,i (ρ) R − − + − W 1/2 yield e D(0,ρ) Leb (W L(f , W 1/2 )) Leb (B). M D (0,ρ) e D(0,ρ) M R R R We conclude by using the estimate (12.384) and the fact that |D ∩ R|, (AA)-case Leb (W ) ≤ −1/10 M q R e D |D ∩ R |, (CC) or (CC*)-case. Proof of estimate (12.371) on the size of the holes. 108 RAPHAËL KRIKORIAN Referring to (12.374)and (8.206)ofProposition 8.1 we see that 1/33 |D| f D h,U where i (ρ) ≤ i ≤ i (ρ).From(7.163) − D + 1/33 1/33 −hN /(33(ln N ) ) i i D D f ≤ ε = e i W i h,U D 1− −N 1− −N i (ρ) e (since i (ρ) ≤ i ) − D hence from (10.302)for any β> 0 (1/ι(ρ))(1−β) |D| exp(−(1/ρ) ). Using (10.303) we then get if ρ 1(D is a disk centered on the real axis) −β 1+τ |D ∩ R| exp(−(1/ρ) ). 12.3. Estimates on the measure of the set of invariant circles: ω Liouvillian, (CC)-Case. — We now assume that (11.335), (11.336)(11.337) hold. Theorem 12.6. — Let ρ = (10A)/(q q ) and assume that q ≥ q .Then, forall n n n+1 n+1 β 1 and n 1 one has 1/4−β m (ρ ) exp(−q )+|(D ∩ R )|. ◦f n n + F n+1 Moreover, 1−β −q (12.388) |(D ∩ R )| e . n + Proof. — The principle of the proof is the same as that of Proposition 12.3 with the HJ ± HJ following modiﬁcations in the notations: we set f = f , f = f and we replace in the i F ± n D n F n D ± − q q proof the indices i (ρ) by i , D, D, Dby D ,D D , i by i , ρ by (4/3)ρ , ρ by 2ρ , ± n n n D n n n n (1/4)− ρ (n) (1/τ )−β U by U and exp(−(1/ρ) ) by exp(−q ). Instead of using the conjugation n+1 relations of Proposition 10.7 (Adapted Normal Forms in the (CC) or (CC*)-case) we use those of Proposition 11.3. Estimate (12.388)isprovedlike(12.371) by noticing that 1/33 1/33 −(1−) f ε ≤ exp(−N ) i W i i h,U D − D n i n and using (11.347). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 109 Remark 12.1. — Note that if the twist condition (11.335)issatisﬁed, then any twist condition TC(A , B) is satisﬁed with A ≥ A. We can thus replace in Theorem 12.6 ρ = (10A)/(q q ) by ρ = (10A )/(q q ) for any ﬁxed A ≥ A(then n has to be chosen n n n+1 n n n+1 larger). 13. Convergent BNF implies small holes 13.1. Case where ω Diophantine in the (AA) of (CC) setting. — We keep here the no- tations of Sections 10 and 12.2, in particular we assume ω is τ -Diophantine and that (10.294), (10.295), (10.296) hold. Lemma 13.1. — If BNF( ◦ f ) converges and is equal to a holomorphic function ∈ O(D(0, 1)) then for all β> 0, ρ 1 and for any D ∈ D β ρ (1/τ )−β (13.389) − exp −(1/ρ) . i (ρ) + D(1/10)D −2 As a corollary, for any D ∈ D and γ ≤ K ρ D HJ (1/τ )−β (13.390) − (·− γ) exp −(1/ρ) . D (4/5)D(1/5)D Proof. — Let us prove inequality (13.389). From (10.333) and Proposition 6.7 one gets (1/τ )−β/2 1−β − b exp(−(1/ρ) ) + exp(−(1/ρ) ) i (ρ) (1/2)D(0,ρ ) (1/τ )−β/2 exp(−(1/ρ) ). (ρ) Since the function − is holomorphic on U and since the triple (U , D i (ρ) i (ρ) + + b −1 −1 (1/10)D, D(0,ρ /2)) is (10b ) | ln ρ| -good, cf. Proposition 10.4,wehavebyDeﬁni- tion 3.3 −1 (1/τ )−β/2 − exp − (10b | ln ρ|) (1/ρ) i (ρ) D(1/10)D τ (1/τ )−β exp(−(1/ρ) ). The inequality (13.390) is then a consequence of (13.389)and (10.334). Corollary 13.2. — If BNF(f ) = then for all β> 0, ρ 1, and any D ∈ D the β ρ radius ρ of the disk D satisﬁes (1/τ )−β ρ exp −(1/ρ) . D 110 RAPHAËL KRIKORIAN Proof. — This results from (13.390) and the Extension Property in Proposition 8.1. Corollary 13.3. — If BNF( ◦ f ) converges then for all β> 0, ρ 1 (recall Nota- F β tion 12.2 for m) (1/τ )−β m (ρ) exp −(1/ρ) . ◦f Proof. — This is a consequence of the previous Corollary 13.2 and of Proposi- 1−2(μ/ι(ρ)) −1 tion 12.3 since #D ρ ρ (cf. (10.309), (10.304), (10.305)). 13.2. Case ω is irrational in the (CC) setting. — The notations here are those of Section 11. In particular we assume that (11.335), (11.336), (11.337) hold. Lemma 13.4. — If BNF( ◦ f ) converges and is equal to ∈ O(D(0, 1)) then for all β 1,n 1 such that q ≥ q β n+1 1−β (13.391) − exp(−q ). D (1/10)D i n+1 n n n −m m/2 As a corollary, for γ q |c | n n n+1 HJ (1/4)−β (13.392) − (·− γ )) exp(−q ). (4/5)D (1/5)D n n n+1 Proof. — Let us prove (13.391). From (11.367) and Proposition 6.7 one gets 1−β/2 − −6 exp(−q ). D(0,q /2) n+1 n+1 (n) (n) Since the function − is holomorphic on U and since the triple (U , D −6 (1/10)D , D(0, q /2)) is 1/(10| ln ρ |)-good (see Proposition 11.2,Item(5)), we have n n n+1 by Deﬁnition 3.3 (remember (11.341)) −1 1−β/2 − exp −(10| ln ρ |) (q ) n n+1 i D (1/10)D n n n 1−β exp(−q ). n+1 The inequality (13.392) is then a consequence of (13.391)and (11.368). Corollary 13.5. — If BNF(f ) = then for any β> 0,n 1 such that q ≥ q ,the β n+1 radius ρ of the disk D satisﬁes q n (1/4)−β ρ exp(−q ). D n+1 n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 111 Proof. — This results from (13.392) and the Extension Property of Proposition 8.1. Corollary 13.6. — If BNF( ◦ f ) converges, then for any β> 0, A ≥ A and n 1 F β,A such that q ≥ q one has n+1 (1/4)−β m (ρ ) exp(−q ), ρ = 10A /(q q ). ◦f n n n+1 n F n+1 Proof. — This follows from the previous Corollary 13.5 and Proposition 12.6 and Remark 12.1. 14. Proof of Theorems C, A and A’ 14.1. Proof of Theorem C. 14.1.1. (AA) Case. — Let f (θ , r) = (θ + ω , r) + (O(r), O (r)) be a real ana- lytic symplectic diffeomorphism of the annulus T ×[−1, 1] satisfying the twist condi- tion (1.14). We can perform some steps of the classical Birkhoff Normal Form proce- dure, Proposition 6.2:for some h > 0,ρ > 0, there exists g = f = id + (O(r), O(r )), 0 Z 10h 10h 2 10h ∈ O (e D(0,ρ )),Z, F ∈ O (e (T × D(0,ρ ))) ∩ O(r ),suchthaton e (T × σ 0 σ h 0 h D(0,ρ )) one has −1 g ◦ f ◦ g = ◦ f , ∀0 ≤ ρ ≤ ρ , F 10h ≤ ρ 0 e (T ×D(0,ρ)) −1 2 3 (2π) (r) = ω r + b (f )r + O(r ) 0 2 j 10 Z(θ , r) = Z (θ )r + r Z (θ , r) j ≥10 j=2 where m is the constant appearing in (10.296). Applying Lemma 2.5 to (r) and 10 3 Lemma 2.2 to r Z (θ , r) we can ﬁnd, for some 0 < ρ ρ ,C Whitney extensions ≥10 0 10h 10h 10h h/10 ∈ O (e D(0, ρ)) and Z ∈ O (e W ) of (, e D(0, ρ)) and (Z, e W ) σ σ h,D(0,ρ) h,D(0,ρ) h/10 such that g := f ∈ Symp (e W ) (see Notations 2.3, 2.6 and 4.8), Z h,D(0,ρ) ex,σ −1 (14.393) ∈ TC(A, B), A = 3min(b (f ), b (f ) ), B ≥ 1 2 2 (14.394) g({r = 0}) = ({r = 0}), g − id 1 ≤ 1/100. Since −1 h/10 g ◦ f ◦ g = ◦ f [e W ] F h,D(0,ρ) 112 RAPHAËL KRIKORIAN one has from (4.86), for any ρ ≤ ρ , L(f , g(W )) = g(L( ◦ f , W )) D(0,ρ) F D(0,ρ) hence, using the fact that g({r = 0}) = ({r = 0}) and g − id ≤ 1/100, we get the inequality (14.395) m (ρ) m (2ρ). f ◦f TheﬁrstpartofTheorem C is then a consequence of Theorem 12.3 applied to ◦ f (which satisﬁes (10.294), (10.295)(10.296)): if we deﬁne D as the set {D, D ∈ D } (each t 2t q q D is associated to a D ∈ D ), formula (1.23) comes from the fact that #D = #D = 2t t 2t #D(U ) (recall the notation (12.370)) and from (10.309), (10.304), (10.305)); on the i (2t) other hand, (1.24) is a consequence of (12.371); ﬁnally (1.25) follows from Theorem 12.3 and inequality (14.395)(we take ρ = t). The second part of Theorem C is a consequence of Corollary 13.2 because if the BNF of f converges, the same is true for that of ◦ f . 14.1.2. (CC) Case. — Let f be a real analytic twist symplectic map of the real disk admitting the origin as an elliptic ﬁxed point with Diophantine frequency ω , (x, y) → 2 2 2 (x, y)+ O (x, y), r(x, y) = (1/2)(x + y ) and satisfying the twist condition (1.14). 2πω r(x,y) We ﬁrst make the symplectic change of variables (4.77) (z,w) = ϕ(x, y), 1 1 √ √ z = (x + iy) x = (z − iw) 2 2 ⇐⇒ i −i √ √ w = (x − iy) y = (z + iw) 2 2 −1 and we write the thus obtained symplectic map (z,w) → f (z,w), f = ϕ ◦ f ◦ ϕ as f = ◦ f , r =−izw. 2πω r F 0 0 We observe that (cf. (4.86)) (14.396) L(f , W) = L(f ,ϕ(W)). Like in the (AA)-case (Section 14.1.1) we perform some steps of Birkhoff Normal Form, Proposition 6.1 and make some Whitney extensions (Lemma 2.2)toobtainfor some h>0, 10h 10h h/10 ρ> 0, maps ∈ O (e D(0, ρ)),F ∈ O (e W ), g ∈ Symp (e W ) sat- σ σ h,D(0,ρ) h,D(0,ρ) ex,σ isfying −1 10h (14.397) g ◦ f ◦ g = ◦ f , [e W ] F h,D(0,ρ) (14.398) g({r = 0}) = ({r = 0}), g − id 1 ≤ 1/100. −1 (14.399) ∈ TC(A, B), A = 3min(b (f ), b (f ) ), B ≥ 1 2 2 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 113 (14.400) ∀ ρ ≤ ρ, F 10h ≤ ρ e W h,D(0,ρ) where m is the constant appearing in (10.296). Applying (14.396), (14.397), (14.398) yields for ρ ≤ ρ (cf. (14.395)) (14.401) m (ρ) ≤ m (2ρ) m (4ρ). f ◦f f F The conclusion of Theorem C is then obtained in the same way as in the previous Sec- tion 14.1.1. 14.2. Proof of Theorem A.— The conclusion of Theorem A is an immediate conse- quence of (1.23), (1.25), (1.26)ofTheorem C:for any β> 0and t 1 (1/τ )−β m (t) exp −(1/t) . 14.3. Proof of Theorem A’.— We proceed like in the previous Section 14.1.2 to obtain (14.397)–(14.400)and then, (14.402) m (ρ) ≤ m (2ρ) m (4ρ). f ◦f f F We now apply Corollary 13.6 to ◦ f . Setting ρ = 10A/(q q ) with A = 3min(b (f ), F n n n+1 2 −1 10 b (f ) ) (cf. (14.399)) we get for any β 1and any n 1such that q ≥ q ,the 2 β n+1 inequality (1/4)−β m (ρ ) exp(−q ). ◦f n n+1 −1 Hence if t := 5min(b (f ), b (f ) )/(q q ) ≤ ρ /4 one has (cf. (14.402)) n 2 2 n n+1 n 1/5 m (t ) ≤ m (2t ) m (4t ) exp(−q ). f n f n ◦f n F n+1 15. Creating hyperbolic periodic points Let ∈ O (D(0, ρ)) satisfy a twist condition (A, B ≥ 1), −1 2 3 (15.403) ∀ r ∈ R, A ≤ (1/2π)∂ (r) ≤ A, and (1/2π)D ≤ B, a ∈ N, a ≥ 10, be the constant appearing in Proposition G.1 of the Appendix and 3 3 −1 (p /q ) the sequence of convergents of ω = (2π) ∂(0). We introduce for n ≥ 1, the n n 0 sequence c deﬁned by −1 (2A) A −1 (15.404) (2π) ∂(c ) = p /q , ≤|c |≤ . n n n n q q q q n n+1 n n+1 114 RAPHAËL KRIKORIAN Proposition 15.1. — Let h > 0,n ∈ N large enough and F ∈ O (T × D(c ,|c | )) such σ h n n that −q h 10 e < |c | F |c | T ×D(c ,|c | ) n h n n and −q h a +1/2 n 3 |F(q , c )|≥ e |c | . n n n Then, −1 2a +1 −4q h 3 n m (c ) ≥ C |c | e . ◦f n n F h The constant C can be chosen to be non increasing w.r.t. h. This proposition will be a consequence of the more precise statement given by the following Proposition 15.2. For p ∈ Z, q ∈ N , p∧ q = 1, p/q small enough, there exists a unique c ∈ D(0, ρ)∩ p/q R such that −1 ω(c ) := (2π) ∂(c ) = p/q. p/q p/q We deﬁne −9 (15.405) ρ = min(|c /4|, q ) p/q p/q and assume that (15.406) ε := F ≤|c | . p/q D(c ,ρ ) p/q p/q p/q 2π −1 ∓iqθ The ±q-th Fourier coefﬁcients of F(·, r), F(±q, r) = (2π) F(θ , r)e dθ satisfy −qh |F(±q, r)| e ε p/q and since F is σ -symmetric, for every r ∈ D(0, ρ) ∩ R, F(q, r) = F(−q, r). Proposition 15.2. — Assume (15.406) is satisﬁed and −qh 10 (15.407) e <ρ p/q −qh (15.408) |F(±q, c )|= ν e ε . p/q q p/q −1 (15.409) ν qρ ≤ 1/q. p/q −1 −qh 3/2 Then, there exists in a neighborhood of T×{c }⊂ T× R an open set of area ≥ C (ν ε e ) , p/q q p/q C > 0, that has an empty intersection with any possible (horizontal) invariant circle of the symplectic diffeomorphism ◦ f . F ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 115 Remark 15.1. — One can choose the constant C to be non increasing with respect to h. Let us see how it provides a proof of Proposition 15.1. Proof of Proposition 15.1.— Since for n large enough −q h −9 10 e < min(|c /4|, q ) −q h 1/2 |F(q , c )|≥ e |c | F 2 n n n T ×D(c ,|c | ) h n n −1/2 −9 lim |c | q min(|c /4|, q ) = 0 n n n n→∞ 1/2 we can apply Proposition 15.2 with q = q , ε =F , ν =|c | , c = c , n p/q T ×D(c ,|c | ) q n p/q n h n n −9 ρ = min(|c /4|, q ).Wethenget p/q n 1/2 −q h 2 m (c ) (|c | e F 2 ) . ◦f n n T ×D(c ,|c | F h n n 2π −1 2 1/2 But, because ((2π) |F(θ , c )| dθ) ≥|F(q , c )|, one has F 2 ≥ n n n T ×D(c ,|c | ) n n 0 h |F(q , c )|,hence n n −1 2a +1 −4q h 3 n m (c ) ≥ C |c | e . ◦f n n The proof of Proposition 15.2 will occupy the next subsections. 15.1. Putting the system into q-resonant Normal Form. — Conditions (15.405)and (15.406) show that we can apply Proposition G.1: it provides us with the following q- resonant Normal Form −1 res g ◦ ◦ f ◦ g = ◦ ◦ f ◦ f cor F RNF 2π(p/q)r F RNF F res (15.410) = − 2π(p/q)r + M (F ) res res res F = F − M (F ) res res −1/q res res F ∈ O (T × D(c , e ρ )), F = F − M (F ); these last two functions are σ h−1/q p/q p/q 0 1/q-periodic (in the θ -variable) and are such that res −1/q F ε p/q T ×D(c ,e ρ ) h−1/q p/q p/q res res F = T (F + O(qρ F )). p/q W N h,D(c ,ρ ) p/q p/q Also, cor −1/4 −1/q (15.411) F exp(−ρ )F , e W h,D(c,ρ) h,D(c,ρ) −2 2 a −5 (15.412) g − id 1 (qρ ) F ≤ ρ . RNF C h,D(c,ρ) 116 RAPHAËL KRIKORIAN −1/q Lemma 15.3. — On T × D(c , e ρ /2) one has 1/q p/q p/q res res res ±iqθ res (15.413)F (θ , r) = u (r) + u (r)e + u (θ , r) 0 1,± ≥2 −1/q where on D(c , e ρ /2) one has p/q p/q res res u (r) = M (F ) = F(0, r) + O(qρ ε ) 0 p/q p/q (15.414) res −qh −qh u (r) = F(±q, r) + O(e qρ ε ) = O(e ε ) p/q p/q p/q 1,± and res −2qh u −1/q e ε . T ×D(c ,e ρ /2) p/q ≥2 1/q p/q p/q Proof. — We recall that from (G.526) q−res res F = T (F + G) q−res (see the notation (G.519)for T )where G −1/q = O(qρ F ). h−1/q,e ρ /2 p/q h,D(c ,ρ /2) p/q p/q p/q Hence −q(h−2/q) −qh (15.415) |G(0, r)| qρ ε , |G(±q, r)| e qρ ε e qρ ε . p/q p/q p/q p/q p/q p/q −2q(h−3/q) −2qh On the other hand since e e q−res ±iqθ −2qh T F − F(0, r) − F(±q, r)e e ε 1/q,ρ /2 p/q N p/q and q−res ±iqθ −2qh T G − G(0, r) − G(±q, r)e −1/q qρ ε e . 1/q,e ρ /2 p/q p/q N p/q Summing these two inequalities and using (15.415)gives (15.413). With these notations res = − 2π(p/q)r + u (r) res res res F = F − u (r). We denote by c ∈ R the point where ∂ (c) = 0; ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 117 res since u ε and satisﬁes the twist condition (15.403) one has D(c ,ρ ) p/q 0 p/q p/q c = c + O(ε ) ∈ D(c,(3/4)ρ ), p/q p/q 2 3 (r) = cst + ( /2)(r − c) + O((r − c) ) −1 res for some A . Since F is σ -symmetric we can write res ±iqθ u (r)e = a(r) cos(qθ) + b(r) sin(qθ) 1,± and from (15.408), (15.414), (15.409) we can assume, shifting the variable θ ∈ T by a translation θ → θ + α (α ∈ T) if necessary, that c c −qh −1 (15.416) b(c) = 0, a(c) = ν e ε , ν = ν − O(qρ ) = ν (1 + o (1)) q p/q q q p/q q q with −qh max(a ,b ) e ε . D(c,ρ ) D(c,ρ ) p/q p/q p/q Thus, res 2 3 (r) = (r) − 2π(p/q)r + u (r) = cst + ( /2)(r − c) + O((r − c) ) res res F (θ , r) = a(r) cos(qθ) + b(r) sin(qθ) + u (θ , r). ≥2 15.2. Coverings. — Like in Section 8.2 (cf. (8.215)) we deﬁne res −2/q res −2/q ∈ O (D(0, qe ρ /2)), F ∈ O (T × D(0, qe ρ /2)) σ p/q σ qh−2 p/q res 2 (15.417) (r) = q (c + r/q) res res 2 F (θ , r) = q F ([θ/q] , c + r/q) mod (2π/q)Z hence res 2 3 2 (r) = cst + r /2 + O(r ) = r /2 + ω(r) res res F (θ , r) = a(r) cos(qθ) + b(r) sin(qθ) + u (θ , r) ≥2 with 2 2 a(r) = q a(c + r/q), b(r) = q b(c + r/q), res 2 res u (θ , r) = q u (θ /q, c + r/q). ≥2 ≥2 Let us deﬁne res res res (15.418) H (θ , r) := (r) + F (θ , r) 2 res = cst + (1/2) r + a(r) cos θ + b(r) sin θ + ω(r) + u (θ , r). ≥2 118 RAPHAËL KRIKORIAN We make explicit the linear plus quadratic part H (θ , r) of (1/2) r + a(r) cos θ + b(r) sin θ at (θ , r) = (0, 0) ∈ R (recall that b(0) = 0) which appears in (1/2) r + a(r) cos θ + b(r) sin θ 2 2 2 = a(0) + r∂ a(0) − (θ /2) a(0) + (r /2)( + ∂ a(0)) + rθ∂ b(0) + g (θ , r) r 0 3 3 where g (θ , r) = O (θ , r) = O(|θ| +|r| ); we can then write 0 3 res (15.419) H (θ , r) = cst + H (θ , r) + ω(r) + g(θ , r) with θ θ 0 θ H (θ , r) = Q , + , ⎨ 2 r r ∂ a(0) r (15.420) − a(0)∂ b(0) ⎪ r Q = ∂ b(0) + ∂ a(0) and res (15.421) g(θ , r) = g (θ , r) + u (θ , r), g (θ , r) = O (θ , r). 0 0 3 ≥2 For further records we mention the following estimates 2 −qh (15.422)max( a ,b ) q e ε D(0,qρ ) D(0,qρ ) p/q p/q p/q 2 −qh (15.423) b(0) = 0, a(0) = q ν e ε , where ν = ν (1 + o (1)) q p/q q q 1/q res 2 −2qh (15.424) u −1/q q e ε T ×D(0,e qρ /2) p/q ≥2 1/q p/q and for (l , l ) ∈ N , l + l ≤ 2and 0 < t <ρ /10 1 2 1 2 p/q l l 2 3−l −l −qh 1 2 1 2 (15.425) ∂ ∂ g (θ , r) q t e ε 0 D(0,t)×D(0,t) p/q θ r l −l l 2 3−l −l −qh l 2 2 −2qh 1 2 1 2 1 (15.426) ∂ ∂ g(θ , r) [q t e ε + q ρ q e ε ]. D(0,t)×D(0,t) p/q p/q θ r p/q 15.3. Existence of a hyperbolic ﬁxed point for f .— We refer to Appendix N for the H + ω deﬁnition of the notion of a (κ, δ)-hyperbolic ﬁxed point. Lemma 15.4. — The afﬁne symplectic map f has a (κ, δ)-hyperbolic ﬁxed point H + ω(r) 5 2 2 (θ , r ) ∈ D(0,ρ ) ∩ R with 0 0 p/q −qh 1/2 δ = κ = q( ν ε e ) (1 + o (1)) p/q 1/q q ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 119 with stable and unstable directions at this point of the form where −qh 1/2 m =±q(ν e ε / ) (1 + o (1)). ± q p/q 1/q Proof. — See Appendix N.2. 15.4. Stable and unstable manifolds of f res . Lemma 15.5. — The symplectic diffeomorphism f res has a (κ, δ)-hyperbolic ﬁxed point 4 2 2 (θ , r ) ∈ D(0,ρ ) ∩ R with 1 1 p/q −qh 1/2 (15.427) κ = δ = q( ν e ε ) (1 + o (1)). q p/q 1/q The stable and unstable directions at this point are of the form where −qh 1/2 m =±q(ν e ε / ) (1 + o (1)). ± q p/q 1/q Proof. — From (15.419), (15.428) f res = f H + ω+g H Q = f ◦ f , g = O (g) H + ω g 1 and from (4.92) of Lemma 4.5,(15.426) (with l + l ≤ 2) and (N.594)weget 1 2 2 5 −qh 4 −2 −2qh Df − id [q ρ e ε + q ρ e ε ] g D(0,10θ )×D(0,10r ) p/q p/q 0 0 p/q p/q 2 5 4 −2 −qh −qh (q ρ + q ρ e )ε e . p/q p/q p/q Because of Lemma 15.4 and (15.428), the Stable Manifold Theorem N.1 of the Appendix shows that the conclusion of the Lemma is true provided for some constant C > 0(cf. (N.589)) −1 f − id 1 ≤ C ρ κδ g C (D(0,10θ )×D(0,10r )) p/q 0 0 a condition that is implied by (recall (15.427) and the fact that from (15.409) one has ν qρ ) q p/q 2 5 4 −2 −qh 2 2 −1 2 (q ρ + q ρ e )< q ρ (< C q ρ ν ). p/q q p/q p/q p/q But (15.405), (15.407) show that this last inequality is satisﬁed if q 1. 120 RAPHAËL KRIKORIAN 15.5. Stable and unstable manifolds of ◦ f . Lemma 15.6. — The diffeomorphism ◦ f has a hyperbolic q-periodic point (θ, r) the local stable and unstable manifolds of which are graphs of C -functions w ,w :]θ − ρ, θ + ρ[→ R such − + that (m/2)|θ − θ|≤|w (θ ) − w (θ )|≤ 2m|θ − θ| (for θ ∈]θ − ρ, θ + ρ[) + − −qh 1/2 m = q(ν e ε / ) (1 + o (1)) q p/q 1/q −1 −qh ρ = C ν e ε . q p/q Proof. — Recall, cf. (15.410), that −1 res (15.429) g ◦ ◦ f ◦ g = ◦ ◦ f ◦ f cor . F RNF 2π(p/q)r F RNF F From (15.417), the pre-image of (θ , r ),by (θ , r) → ([(θ − α )/q] , c + r/q) is 1 1 c mod (2π/q)Z 4 4 a q-periodic orbit O ⊂ T×]c − ρ , c + ρ [⊂ T×]c − ρ /3, c + ρ /3[ of q p/q p/q p/q p/q p/q p/q p/q p/q res res res ◦ f as well as of ◦ ◦ f (F is 2π/q-periodic); Lemma 15.5 tells us that 2π(p/q)r F F 4 4 this periodic orbit is hyperbolic. Let u ∈ T×]c − ρ , c + ρ [ be a point of O and 0 p/q p/q q p/q p/q res denote ϕ = ◦ ◦ f . One has ϕ (u ) = u and we want to ﬁnd a hyperbolic 2π(p/q)r 0 0 cor cor ﬁxed point for (ϕ ◦ f ) (the q-th iterate of ϕ ◦ f )close to u . F F 0 We can write q q (ϕ ◦ f cor ) = ϕ ◦ j where −(q−1) q−1 −1 cor cor cor j = (ϕ ◦ f ◦ ϕ )◦···◦ (ϕ ◦ f ◦ ϕ) ◦ f . F F F Since ( ◦ ) 2 1 uniformly in n and 2π(p/q)r C (T×]c −ρ /3,c +ρ /3[) p/q p/q p/q p/q res −2 F 2 ε ρ 1 C (T×]c −ρ /3,c +ρ /3[) p/q p/q p/q p/q p/q p/q one has for n ≤ ρ /ε , p/q p/q (15.430) ϕ 1 C (T×]c −ρ /3,c +ρ /3[) p/q p/q p/q p/q −1 and consequently (q ε ) cor j − id 1 qF 1 C (T×]c −ρ /3,c +ρ /3[) C (T×]c −ρ /3,c +ρ /3[) p/q p/q p/q p/q p/q p/q p/q p/q −1/3 exp(−ρ ) p/q where we have used (15.411). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 121 q q −1 −1 Replacing ϕ and j by T ◦ ϕ ◦ T and T ◦ j ◦ T where T : u → u − u we can 4 2 2 assume that u = 0 ∈ D ( c,ρ ) ⊂ D ( c,ρ /3) ⊂ T×] c −|c /3|, c + ρ /3[.We 0 R R p/q p/q p/q p/q q q then have ϕ (0) = 0and the matrixDϕ (0) is (κ, δ)-hyperbolic with 2 −qh δκ = q ν e ε (1 + o (1)). q p/q 1/q q q Write ϕ (u) = Dϕ (0)ξ (u) with ξ(0) = 0, Dξ(0) = id so that q q ϕ ◦ j = Dϕ (0) ◦ ξ ◦ j. Observe that for 0 <ρ <ρ /4and k = 0, 1 p/q k k D (ξ ◦ j − id) 0 2 D (ξ − id) 0 2 +j − id 1 2 C (D (0,ρ)) C (D (0,ρ)) C (D (0,ρ)) R R R −1/3 2−k ρ + q exp(−ρ ). p/q Let us choose −1 −qh ρ = C ν e ε q p/q with C large enough. The Stable Manifold Theorem (cf. Appendix, Theorem N.1)shows that the diffeomorphism ϕ ◦ j has a hyperbolic ﬁxed point the stable and unstable man- ifolds of which are graphs of C functions of the form w , w :] − ρ/2,ρ/2[→ R, − + w < 0 < w ,suchthatfor all θ ∈] − ρ/2,ρ/2[ one has − + (3/2)m θ ≤ w (θ ) ≤ (2/3)m θ ≤ 0 ≤ (2/3)m θ< w (θ ) ≤ (3/2)m θ. − − − + + + −1 To conclude the proof of the Lemma we set w = w ◦ g and note that ± ± RNF −1 q g ◦ ◦ f ◦ g = ϕ ◦ j F RNF RNF −1 with g − id ≤ 1/10 (see (15.412)). RNF 15.6. End of the proof of Proposition 15.2.— Let V be the set V={(θ , r), θ ∈[θ, θ + ρ/2],w (θ ) ≤ r ≤ w (θ )} − + the boundary of which is made by two pieces of stable and unstable manifolds and the vertical segment L := {θ + ρ/2}×[w (θ + ρ/2), w (θ + ρ/2]. By a theorem of Birkhoff − + [4](cf. also [21]), any invariant (horizontal) curve of the twist diffeomorphism ◦ f is the graph of a Lipschitz function γ : T →[−1, 1]; if this curve intersects the stable or unstable manifold of (θ, r) it must be included in the union of these stable and unstable manifolds which is impossible. So if this invariant curve intersects the interior of V it has to enter in V by ﬁrst entering the vertical segment L by the right. But this is clearly impossible also (see Figure 11). 122 RAPHAËL KRIKORIAN γ(·) w (·) w (·) (θ, r) V V θ + ρ/2 FIG. 11. — Invariant graphs cannot intersect the interior of V Now the domain V has an area which is area(V) ρ × (m − m ) + − −qh 3/2 (ν e ε ) . q p/q This concludes the proof of Proposition 15.2 if we notice that the dependence on h of the implicit constant in the symbol appears only when we apply Proposition G.1 on Resonant Normal Forms (cf. Remark G.1). 16. Divergent BNF: proof of Theorems E, B and B’ We now use the result of the previous Section to construct examples of real analytic symplectic diffeomorphisms of the disk and the annulus with divergent BNF. 16.1. Proof of Theorems E and B: the (AA) Case. — Let f = ◦ f be a real 2πω r O(r ) analytic symplectic twist map of the annulus of the form (1.6) and satisfying the twist condition (1.14). We perform a Birkhoff Normal Form, cf. Proposition 6.2,on f up to order a ,where a is the integer of Proposition G.1 of the Appendix that appears in 3 3 Proposition 15.1: there exist ρ> 0, g ∈ Symp (T × D(0, ρ)) exact symplectic, ∈ ex,σ O (D(0, ρ)),F ∈ O (T × D(0, ρ)) such that σ σ h −1 g ◦ f ◦ g = ◦ f where (b = 0) −1 2 3 a 2 (2π) (r) = ω r + b r + O(r ), F(θ , r) = O(r ), g − id = O(r ). 0 2 −1 Note that for ρ small enough satisﬁes a ((5/2) min(b , b ), B)-twist condition on D(0, ρ) (B ≥ 1). In particular, if (p /q ) are the convergents of ω and c ∈ R (n large n n n≥1 0 n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 123 enough) is the point where −1 (5/2) max(b (f ), b (f ) ) 2 2 −1 (16.431) (2π) ∂(c ) = p /q , |c |≤ n n n n q q n n+1 (cf. (15.404) and the twist condition satisﬁed by ) one has F 2 |c | . T ×D(c ,|c | ) n h n n For (ζ ) ,(ζ ) ∈[−1, 1] ,let G ∈ T × D(0, 1) deﬁned by 1,k k≥1 2,k k≥1 ζ h a −q h −q h 3 k k G (θ , r) = r ζ e cos(q θ) + ζ e sin(q θ). ζ 1,k k 2,k k k≥1 We now deﬁne f ∈ Symp (T × D(0, 1)),F ∈ O (T × D(0, ρ)) by ζ ζ σ h f = f ◦ f , ζ G −1 −1 ◦ f := g ◦ f ◦ g = ◦ f ◦ g ◦ f ◦ g. F ζ F G ζ ζ Lemma 16.1. — For n, c as above, there exists a set J (F) ⊂[−1, 1] of 2-dimensional n n 1/2 2 N 2 Lebesgue measure |c | such that for any ζ ∈ ([−1, 1] ) , such that ζ ∈[−1, 1] J (F) one n n n has 2a +1 −4q h 3 n m (c ) |c | e . ◦f n n Proof. — Let α ∈ T and ν ≥ 0be such that n q iq θ −iq θ a −q h n n 3 n F(q , c )e + F(−q , c )e =|c | ν e cos(q θ + α ). n n n n n q n q n n −1 a 2 Since ◦ f = ◦ f ◦ g ◦ f ◦ g,F, G = O(r ) and g − id = O(r ) we see that F F G ζ ζ ζ a +1 (16.432)F = F + G + O(r ). ζ ζ We now assume c > 0 for simpler notations (the case c < 0 istreated in thesameway). n n We can write a −q h −q h 3 k k G (θ , r) = r ζ e cos(q θ + α ) + ζ e sin(q θ + α ) ζ 1,k k q 2,k k q k k k≥1 −iα with ζ − iζ = e (ζ − iζ ) and from (16.432) we see that 1,k 2,k 1,k 2,k iq θ −iq θ n n F (q , c )e + F (−q , c )e ζ n n ζ n n a −q h 3 n =|c | e (ν +ζ +u (ζ )) cos(q θ+α )+(ζ +v (ζ )) sin(q θ+α ) n q 1,n n n q 2,n n n q n n n 124 RAPHAËL KRIKORIAN where sup (|u (ζ )|,|v (ζ )|) |c |. n n n 2 N ζ∈([−1,1] ) We can thus write for ζ ,ζ ∈] − 1, 1[ 1 2 a −q h 3 n 2|F (q , c )|= ν (ζ )|c | e ζ n n n n with 2 2 2 ν (ζ ) = (ν + ζ + u (ζ )) + (ζ + v (ζ )) n q 1,n n 2,n n ≥|ζ − O(c )| . 2,n n −iα 2 Since ζ − iζ = e (ζ − iζ ), one can hence ﬁnd a set J (F) ⊂[−1, 1] of 2- 1,k 2,k 1,k 2,k n dimensional Lebesgue measure 1/2 |J (F)| |c | n n such that 2 1/2 (ζ ,ζ )∈[−1, 1] J (F) %⇒ |ν (ζ )| |c | . 1,n 2,n n n n By Proposition 15.1 we thus have 2a +1 −4q h 3 n m (c ) |c | e . ◦f n n 2 N Lemma 16.2. — Let N ⊂ N be inﬁnite. Then, for almost every ζ ∈ ([−1, 1] ) for the ⊗N product measure μ = (Leb 2 ) , there exists an inﬁnite subset N ⊂ N such that for all n ∈ N ∞ [−1,1] 2a +1 −4q h 3 n m (c ) |c | e . ◦f n n Proof. — Since the random variables ζ , n ∈ N are independent, for any m ∈ N, the event {ζ ∈ J (F), ∀ n ≥ m} has zero μ -probability as well as their union. Hence for n n ∞ μ -almost every ζ ∈ X , one has for inﬁnitely many n ∈ N , ζ ∈ / J (F) and we conclude ∞ n n by Lemma (16.1). 16.1.1. Proof of Theorems E and B.— We now observe that if ω is Diophantine with exponent τ ln q n+1 τ = lim sup , ln q then for any β> 0 there exists a inﬁnite set N such that for all n ∈ N β β τ−β/4 q ≥ q . n+1 n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 125 On the other hand p 1 1 |c | ω − n 0 1+τ−β/4 q q q n n n+1 n hence (1/(1+τ))+β/4 q (1/|c |) n n and consequently, from Lemma 16.2, for an inﬁnite number of n ∈ N ( )+β/2 1+τ 2a +1 −4q h (16.433) m (c ) |c | e exp − . ◦f n n |c | We observe that since t ≥ 2|c | (cf. (1.20)and (16.431)) one has n n ( )+β 1+τ (16.434) m (t ) m (c ) exp − . f n ◦f n ζ F |t | If β is chosen so that 1 1 + β< − β, 1 + τ τ the estimate (16.434), when compared to the conclusion of Theorem A, shows that the 2 N Birkhoff Normal Form of f is divergent for μ -almost every ζ ∈ ([−1, 1] ) . ζ ∞ This concludes the proof of Theorem E and, as a consequence, of that of Theo- rem B (in the (AA) case). 16.2. Proof of Theorems E’, B and B’: (CC) Case. — Let f be a real analytic symplectic diffeomorphism of the disk admitting the origin 0 as an elliptic equilibrium with irrational frequency ω and satisfying the twist condition (1.14); we assume that it is of the form 2 2 a 2 2 f = + O((x + y ) ) ((1/2)(x +y )) with ∈ O (D(0, 1)). Passing to the (z,w)-variables (cf. (4.77)) we can write −1 ϕ ◦ f ◦ ϕ = ◦ f where F ∈ O (D(0, 1) ) a a 3 3 F(z,w) = O((zw) )(and not only O (z, w)). Let as before (p /q ) be the convergents of ω and c ∈ R the point where n n n≥1 0 n −1 (2π) ∂(c ) = p /q (cf. (15.404)). n n n 126 RAPHAËL KRIKORIAN 2 N 2 For (ζ ) ∗ ∈ ([−1, 1] ) ,let G ∈ O (D(0, 1) ) n n∈N ζ σ 1,k a −1/2 q −1/2 q 3 k k G (z,w) = (−izw) × ((i z) + (i w) ) k=1 2,k −1/2 q −1/2 q k k + × ((i z) − (i w) ). 2i We now deﬁne (recall the deﬁnition of G in Section 1.4.3) ◦ f = ◦ f ◦ F F G ζ ζ −1 f = ϕ ◦ ( ◦ f ) ◦ ϕ = f ◦ = f ◦ .(cf .(4.82)) ζ F G ◦ϕ G ζ ζ Lemma 16.3. — Assume that for some n large enough, c is positive. Then, there exists J (F) ⊂ n n 2 1/2 [−1, 1] of Lebesgue measure c such that if ζ = (ζ ,ζ )/∈ J (F) one has n 1,n 2,n n 2a +1 −4q h 3 n m (2c ) m (c ) c e . f n ◦f n ζ F Proof. — We deﬁne h =−(1/2) ln(c + c ), n n n 1 and since |ω − | c one has 0 n q q q n n n+1 h = (−1/2) ln c + O(c ) n n n = (−1/2) ln c − O(1/q ) hence −q h q /2 O(1/q ) 10 n n n n (16.435) e = c e < c . n n CC CC 2 h 2 1/2 Let W = W ={(z,w) ∈ C , max(|z|,|w|) ≤ e (c + c ) , −izw ∈ 2 n n n h ,D(c ,c ) n n D(c , c )}. One has a a 3 3 CC CC (16.436) F |c | , G c . W n ζ W n n n Using Lemma K.1 we can pass to (AA)-coordinates: if ψ is the diffeomorphism deﬁned in (4.79) −1 CC AA 2 ψ (W ) ⊃ W = T × D(c , c ) 2 2 h n − h ,D(c ,c ) h ,D(c ,c ) n n n n n n n AA AA 2 and we can introduce F , F ∈ O (T × D(c , c )) (cf. (4.82)) σ h n ζ n −1 −1 ψ ◦ f ◦ ψ = f AA,ψ ◦ f ◦ ψ = f AA = f AA ◦ . F − F F − F F G ◦ψ − − ζ ζ − ζ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 127 AA AA AA AA Since F = F ◦ ψ + O (F) and F = F + G ◦ ψ + O (F , G ◦ ψ ) (cf. (4.94), − 2 ζ − 2 ζ − AA (4.92)) one has on W AA a AA AA (3/2)a 3 3 (16.437) F AA c , F = F + G ◦ ψ + O(c ). W ζ − n n ζ n ζ If we deﬁne ν and α ∈ T by n n AA iq θ AA −iq θ a −q h n n 3 n n F (q , c )e + F (−q , c )e =|c | ν e cos(q θ + α ) n n n n n q n q n n we see that on T × D(c , c /2) (cf. (16.437)) h −1 n n n AA AA a q /2 (3/2)a 3 k 3 F = F (θ , r) + r r (ζ cos(q θ) + ζ sin(q θ)) + O(c ). 1,k k 2,k k ζ n k=1 Hence AA iq θ −iq θ n n F (q , c )e + F (−q , c )e n n ζ n n a (1/2)a −q h q /2 3 3 n n n = c (ν + O(c ))e + c ζ cos(q θ + α ) q 1,n n q n n n n n q /2 (1/2)a −q h n 3 n n + c ζ + O(c )e sin(q θ + α ) 2,n n q n n −iα with ζ − iζ = e (ζ − iζ ).Wethushave(cf. (16.435)) 1,k 2,k 1,k 2,k AA a q /2 (1/2)a −q h 3 n 3 n n 2|F (q , c )|≥ c |c ζ + O(c )e | n n 2,n ζ n n n a −q h O(1/q ) (1/2)a −q h 3 n n n 3 n n ≥ c |e e ζ + O(c )e | 2,n n n a −q h (1/2)a 3 n n 3 c e |ζ + O(c )| 2,n n n and we see that if 1/2 |ζ |≥ c 2,n one can apply Proposition 15.1 (cf. (16.435)): −1 2a +1 −4q h 2a +1 −4q h 3 n 3 n m (c ) C c e c e . ◦f n AA h n n Now, since c is positive and m (2c ) m (c ) this provides n f n ◦F n ζ ζ 2a +1 −4q h 3 n (2c ) m (c ) c e . f n ◦f ◦ n ζ F G We can deduce the analogue of Lemma 16.2 128 RAPHAËL KRIKORIAN Lemma 16.4. — Let N be an inﬁnite set of n ∈ N for which c > 0. Then, for almost every 2 N ζ ∈ ([−1, 1] ) , there exists an inﬁnite subset N ⊂ N such that for all n ∈ N 2a +1 −4q h 3 n m (2c ) m (c ) c e . f n ◦f n ζ F 16.2.1. Proof of Theorems E’ and B (CC) Case, ω Diophantine. — We want to apply thepreviousLemma 16.4 to an inﬁnite set N such that for all n ∈ N one has both τ− (16.438) c >0and q ≥ q . n n+1 Such a set may not exist for arbitrary choices of ω (Diophantine) and . On the other hand, if one chooses the sign of ∂ (0) depending on ω (or more precisely its sequence of convergents) this is possible. Let β> 0and τ−β/2 N ={n ∈ N, q ≥ q }, Q ={p /q , n ∈ N }. β n+1 β n n β Since N is inﬁnite, one of the two sets Q = Q ∩ (]ω ,±∞[) is inﬁnite. We deﬁne β β 0 s (ω ) as the non-empty subset of {−1, 1} such that ±1 ∈ s (ω ) if and only if Q is β 0 β 0 inﬁnite. −1 We now assume that ω(r) := (2π) ∂(r) is of the form ω(r) = ω + 2b r + O(r ), with sign(b ) ∈ s (ω ). 0 2 2 β 0 For the sake of simplicity we shall assume that 1 ∈ s(ω ) and b > 0(the case −1 ∈ s(ω ) 0 2 0 and b < 0 is treated similarly). The sets + −1 + N := N ={n ∈ N , p /q >ω }, C = ω (Q ) β n n 0 β β β β are then inﬁnite; note that C ⊂]0,∞[ and its points c , n ∈ N accumulate zero. β n We then apply Lemma 16.4: for almost every ζ and inﬁnitely many n ∈ N 2a +1 −4q h 3 n m (2c ) c e f n ζ n and, arguing like in Subsection 16.1.1, we see that setting t = 2c we have for inﬁnitely n n many n ∈ N = N ( )+β 1+τ m (t ) exp − . f n |t | 16.2.2. Proof of Theorems E’ and B’: (CC) Case, ω Liouvillian. — Since ω is Liouvil- 0 0 lian, there exists an inﬁnite set N ⊂ N such that ln q n+1 lim =∞. n∈N ln q n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 129 We deﬁne Q ={p /q , n ∈ N , p /q ∈]ω ,±∞[} n n n n 0 + − and s(ω ) as the non-empty subset of {−1, 1} (one of the two sets Q , Q is inﬁnite) such that ±1 ∈ s(ω ) if and only if Q is inﬁnite. We assume that 1 ∈ s(ω ) and b > 0(the case −1 ∈ s(ω ) and b < 0is treated 0 2 0 2 + −1 + similarly) and we set N ={n ∈ N , p /q >ω }. The set C = ω (Q ) is inﬁnite, con- n n 0 tained in ]0,∞[ and its points c , n ∈ N accumulate 0. We now apply Lemma 16.4: for almost every ζ one has for inﬁnitely many n ∈ N 2a +1 −4q h 3 n m (2c ) c e . f n ζ n For these n’s one has p 1 c |ω − | , n 0 q q q n n n+1 and, for any β> 0, provided n is large enough, β/2 β 2a +1 −4q h 2(2a +1) 3 n 3 c e ( ) exp(−q ) exp(−q ). n+1 n+1 n+1 If we set t = 2c (cf. (16.431)) n n −1 5(b + b ) 2c ≤ t := n n q q n n+1 hence m (t ) ≥ m (2c ) exp(−q ) f n f n ζ ζ n+1 for inﬁnitely many n in N := N . Acknowledgements The author wishes to thank Alain Chenciner and Håkan Eliasson for their con- tinuous encouragements, Abed Bounemoura, Bassam Fayad, Jacques Féjoz, Jean-Pierre Marco, Stefano Marmi, Laurent Niederman, Ricardo Pérez-Marco, Laurent Stolovitch for interesting discussions and all the participants of the Groupe de travail de Jussieu for their patient listening of this work and their constructive comments. The author is also very grateful to the referees the comments of which helped improving signiﬁcantly this text. He is in particular indebted to one of the referees who pointed out an error in the exponent of Theorems A and C in the submitted version of this paper. 130 RAPHAËL KRIKORIAN Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. 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Appendix A: Estimates on composition and inversion A.1 Proof of Lemma 4.4.— We shall do the proof in the (AA)-case; the proof in the (CC)-case follows the same lines. −δ We can assume that e W = W −δ is not empty (otherwise, there is nothing h,U h−δ/2,e U to prove). By (2.53), there exists a numerical constant C > 0such that if δ> 0satisﬁes −2 −1 (A.439)CF δ d(W ) < 1, W h,U h,U −δ then for any ﬁxed θ ∈ T and any ﬁxed r ∈ e U, the map U → U, R → r − ∂ F(θ , R) h−2δ θ is contracting and by the Contraction Mapping Principle there thus exists a unique R ∈ U −δ depending holomorphically on (θ , r) ∈ T × e Usuch that h−δ/2 r = R + ∂ F(θ , R). On the other hand, assumption (A.439) and Cauchy’s inequality (2.53) show that if C is large enough −1 F(θ , R)| δ ×F <(1/2)δ, |∂ R h,U hence ϕ := θ + ∂ F(θ , R) ∈ T . R h−δ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 131 We can thus deﬁne a holomorphic map −δ f : T × e U → T × U F h−δ/2 h−δ by r = R + ∂ F(θ , R) (A.440) f (θ , r) = (ϕ, R) ⇐⇒ ϕ = θ + ∂ F(θ , R). Notice that the maps (θ , r) → ϕ(θ, r)− θ , (θ , r) → R(θ , r)− r such deﬁned are Lipschitz −2 −2 with Lipschitz constant δ d(U) F . Thus, if for some numerical constant large h,U enough −2 −2 (A.441)Cδ d(U) F < 1, h,U −δ the map f is a holomorphic diffeomorphism from T × e U onto its image. F h−δ/2 −2δ Conversely, if (A.439)issatisﬁed, given (ϕ, R) ∈ T × e U, thesamearguments h−δ −δ as those developed above show there exists a unique (θ , r) ∈ T × e Usuch that h−δ/2 f (θ , r) = (ϕ, R).Wethushaveif(A.441)issatisﬁed −2δ −δ T × e U ⊂ f (T × e U). h−δ F h−δ/2 Finally, we observe that the diffeomorphism f is exact symplectic which means that the differential form Rdϕ − rdθ is exact; in particular, it is symplectic. Indeed Rdϕ − rdθ =−ϕd R + d(ϕR) − rdθ =−(θ + ∂ F(θ , R))d R − (R + ∂ F(θ , R))dθ + d(ϕR) R θ =−d F + d(ϕR) − d(θ R) = d(−F + (ϕ − θ)R) (observe that the function −F + (ϕ − θ)R =−F(θ , R) + ∂ F(θ , R)R is well deﬁned on T × U). We have thus proven that there exists a numerical constant C > 0such that if −2 −2 (A.442) Cδ d(U) F < 1 h,U the diffeomorphism f previously deﬁned is exact symplectic and −2δ −δ (A.443) e W ⊂ f (e W ) ⊂ W . h,U F h,U h,U Estimate (4.91) comes from (A.440)and max|∂ F(θ , R) − ∂ F(ϕ, r)|| ≤ 2D FDF. i i i=1,2 132 RAPHAËL KRIKORIAN A.2 Proof of Lemma 4.5.— We illustrate the proof in the (AA)-Case (it is the same in the (CC)-case). Since f is close to the identity, the map f : (θ , R) → (ϕ, r) ⇐⇒ f (θ , r) = (ϕ, R) deﬁnes a diffeomorphism such that f − id = O(f − id) and since f is exact symplectic we know (cf. Section A.1)that ϕd R + rdθ = d F for some holomorphic function F : (θ , R) → F(θ , R). Since F(θ , R) = (ϕd R + rdθ) where γ is a path joining (0, R )∈{θ ∈ C, θ,R 0 θ,R |$θ| < h}× Uto (θ , R), the function F, which is unique up to the addition of a constant, thus satisﬁes F = O(f − id). The estimate (4.92) is a consequence of (4.87) and the fact that |∂ F(θ , R) − ∂ F(θ , r)|≤D∂ F|R − r|≤D∂ F∂ F θ θ θ θ θ |∂ F(θ , R) − ∂ F(θ , r)|≤D∂ F|ϕ − θ|≤D∂ F∂ F. R R R R R A.3 Proof of Lemma 4.6. — 1) Proof of (4.95). One has r = R + ∂ F(θ , R) f (θ , r) = (ϕ, R) ⇐⇒ ϕ = θ + ∂ F(θ , R) R = Q f (ϕ, R) = (ψ, Q) ⇐⇒ ψ = ϕ + ∂ (Q) hence Q = Rand ψ = ϕ + ∂ (R) = θ + ∂ F(θ , R) + ∂ (R) R R = θ + ∂ ( + F)(θ , Q) thus, since r = R+ ∂ F(θ , R) = Q+ ∂ F(θ , Q) and does not depend on the θ -variable, θ θ one has r = Q + ∂ ( + F)(θ , Q) ψ = θ + ∂ ( + F)(θ , Q) which is equivalent to f (θ , r) = (ψ, Q) = f ◦ f (θ , r). +F F 2) Proof of (4.93). Assume that f (θ , r) = (ϕ, R) and f (ϕ, R) = (ψ, Q).Then F G r = R + ∂ F(θ , R) (A.444) f (θ , r) = (ϕ, R), ϕ = θ + ∂ F(θ , R) R ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 133 R = Q + ∂ G(ϕ, Q) (A.445) f (ϕ, R) = (ψ, Q), ψ = ϕ + ∂ G(ϕ, Q) Qdψ − rdθ = Qdψ − Rdϕ + Rdϕ − rdθ = d(−F − G + (ϕ − θ)R + (ψ − ϕ)Q). If f ◦ f = f then one has Qdψ − rdθ = d(−H + Q(ψ − θ)) and then G F H 0 = d(−H + F + G + Q(ψ − θ) − R(ϕ − θ) − Q(ψ − ϕ)) = d(−H + F + G − (Q − R)(ϕ − θ)) and so H(θ , Q) = cst + F(θ , R) + G(ϕ, Q) − (Q − R)(ϕ − θ). Let us write H(θ , Q) = F(θ , Q) + G(θ , Q) + A(θ , Q) where −A = F(θ , Q) − F(θ , R) + G(θ , Q) − G(ϕ, Q) + (Q − R)(ϕ − θ) = F(θ , Q) − F(θ , R) + G(θ , Q) − G(ϕ, Q) − ∂ G(ϕ, Q)∂ F(θ , R) ϕ R We can now estimate A ≤∂ F Q − R +∂ G ϕ − θ h−δ,U R h,U h,U ϕ h,U h,U +∂ G(ϕ, Q) ∂ F(θ , R) ϕ h,U R h,U ≤∂ F ∂ G +∂ G ∂ F R h,U ϕ h,U ϕ h,U R +∂ G ∂ F ϕ h,U R h,U and deduce (4.93). 3) Proof of (4.94). We just write −1 f ◦ f = f ◦ f 2 F+G F+G −G+O(|D G|DG|) = f (using (4.93)) F+DFO (G) −1 and a similar expression for f ◦ f = f 2 . F+G −F+O(|D F||DF|) The proof of (4.93)and (4.94) is the same in the (CC)-case. A.4 Proof of Proposition 4.7.— We ﬁrst state two lemmata. Lemma A.1. — Let W be an open subset of M = C or T × C, v ∈ O(W) and g − id ∈ O(W) such that g − id 1. Then if v is small enough W W −1 (A.446) (id + v) ◦ g ◦ (id + v) = g ◦ (id +[g]· v + O (v)) 2 134 RAPHAËL KRIKORIAN where −1 (A.447) [g]· v =−v + (Dg · v) ◦ g. Proof. — One has −1 (id + v) ◦ g ◦ (id + v) = g ◦ (id − v + O (v)) + v ◦ g ◦ (id − v + O (v)) 2 2 = g − Dg · v + v ◦ g + O (v) hence −1 −1 g ◦ (id + v) ◦ g ◦ (id + v) −1 = g ◦ g − Dg · v + v ◦ g + O (v) −1 −1 = id − Dg ◦ g · Dg · v + Dg ◦ g · v ◦ g + O (v) −1 = id − v + (Dg · v) ◦ g + O (v). Lemma A.2. — If ∈ O(U), Y ∈ O(W ∪ (W ))) then h,U h,U −1 (A.448) f ◦ ◦ f = ◦ f Y []·Y+O (Y) where []· Y = Y ◦ − Y. Proof. — From Lemma A.1,and (4.92)wehave −1 (A.449) f ◦ ◦ f = ◦ (id +[ ]· (J∇ Y) + O (Y)) Y 2 −1 = ◦ id − J∇ Y + (D · (J∇ Y)) ◦ + O (Y) . On the other hand (A.450)J∇(Y ◦ ) = J ( D ) · (∇ Y ◦ ). t −1 Because is symplectic, one has J (D ) = (D ) J, and we deduce from both (A.449), (A.450)that −1 f ◦ ◦ f = ◦ (id + J∇ Y − J∇(Y ◦ ) + O (Y)) Y 2 = ◦ f []·Y+O (Y) where []· Y = Y ◦ − Y. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 135 From (4.92), (4.93)(we usethe fact that D(O(DFG))=DFO (G)) −1 (A.451) f ◦ f ◦ f = f Y F F+DFO (Y) Y 1 and on the other hand, from Lemma A.2 −1 (A.452) f ◦ ◦ f = ◦ f . Y []·Y+O (Y) Y 2 Hence (we use (4.93) in the last line of the following equations) −1 −1 −1 (A.453) f ◦ ◦ f ◦ f = f ◦ ◦ f ◦ f ◦ f ◦ f Y F Y Y F Y Y Y (A.454) = ◦ f ◦ f []·Y+O (Y) F+DFO (Y) 2 1 (A.455) = ◦ f , F+[]·Y+DFO (Y) and the conclusion follows if F = F+[]· Y+DFO (Y). Appendix B: Whitney type extensions B.1 Proof of Lemma 2.2.— Let χ : R→[0, 1] be a smooth function with support −δ/2 −δ/2 in [−1, 1] and equal to 1 on [−e , e ] such that j −j (B.456)sup|∂ χ | δ . δ/2 2 2 −δ We deﬁne for r ∈ C, η(r) = χ ((e |r|) /ρ ) and for i ∈ J , η (r) = (1 − χ ((e |r − δ U i δ 2 2 −δ −δ/2 c |) /ρ )).Notethat η is equal to 1 on e D(0,ρ) and 0 on C e D(0,ρ) and η is i i δ δ/2 equal to 1 on C e D(c ,ρ ) and 0 on e D(c ,ρ ) hence ζ = η η is equal to 1 on i i i i i i∈J −δ −δ/2 δ/2 e Uand 0 onV := (C e D(0,ρ)) ∪ e D(c ,ρ ). The union of the open sets i i i∈J −1/10 W −δ/10 (resp. e U) and W (resp. V) is W (resp. C) and on their intersection the h,e U h,V h,C functions ζ F and 0 coincide. As a consequence, one can extend ζFby 0 on W as a h,V Wh smooth function F : W → W . Note that since ζ is σ -symmetric, the same is true h,C h,C Wh Wh −δ for F and that F and F coincide on W −δ (which contains e W ). h,e U h,U Wh To get the estimates on the derivatives of F we observe from (B.456)and the deﬁnitions of η, η that j j −k −2k −2k max(max sup|D η|, max sup|D η |) δ max(ρ ,ρ ) i 0≤j≤k 0≤j≤k i C C and since max (η, η ) ≤ 1, one has by Leibniz formula i i j k −k −2k −2k max sup|D ζ| (#J + 1) δ max(ρ ,ρ ). 0≤j≤k i Wh Hence F := ζ Fsatisﬁes j Wh k −2k j sup D F ≤ C(1 + #J ) (δ d(U)) maxD F . W U W h,C −δ/10 h,e U 0≤j≤k 0≤j≤k 136 RAPHAËL KRIKORIAN −n −1 n B.2 Proof of Lemma 2.5.— Write (2π) (z) = b z with |b |≤ ρ , n n n=0 0 −1 2 −1 n (2π) (z) = b + b z + b z , (2π) (z) = b z .For 0 ≤ j ≤ 3and δ> 0, 2 0 1 2 ≥3 n n=3 there exists C > 0such that for any ρ ≤ ρ /2 j 0 j 3−j (B.457) D ≤ C ρ . ≥3 D(0,ρ) j Let χ : C →[0, 1] be a real symmetric smooth function with support in D(0, 1) and equal to 1 on D(0, 1/2). We deﬁne the real symmetric function deﬁned on C Wh (z) = (z) + χ(z/ρ) (z). 2 ≥3 Wh For any z ∈ D(0,ρ/2) one has (z) = (z) and by (B.457) and Leibniz formula, for j j some constant B depending only on b , b , b , D χ 0 , D ,0 ≤ j ≤ 3, one 0 1 2 C ≥3 D(0,ρ ) has 3 Wh ∀z ∈ C, D (z) ≤ B. 2π j j On the other hand, for some constant C depending only on, ∂ χ 0 , ∂ ,0 ≤ C D(0,ρ ) j ≤ 2 2 Wh ∀ t ∈ R, ∂ (t) − 2b ≤ Cρ 2π and if ρ = ρ is chosen small enough so that Cρ< b , one has (we assume b > 0) b ≤ 2 2 2 2 Wh ∂ (t) ≤ 3b . 2π B.3 Proof of Proposition 2.7.— The proof will follow from the following two lemmas. Lemma B.1. — Let β ∈ R, ν> 0;iffor some t + is ∈ U (t, s ∈ R)one has |ω(t + is) − β| <ν, then |ω(t) − β|≤ (7/6)ν (B.458) |s|≤ (4/3)Aν. Proof. — Since ω is holomorphic on U one has for any z ∈ U, ∂ω(z) = 0(we use in this proof the usual notations ∂ = (1/2)(∂ + i∂ ) and ∂ = (1/2)(∂ − i∂ )). For any point t s t s z ∈ D(0,ρ) one has (cf. Lemma 2.1) dist(z, U) ≤ 2a(U) ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 137 and from the fact that D∂ω≤ B we thus get using condition (2.60) (B.459) ∂ω 0 ≤ a(U)B C (D(0,ρ)) −1 ≤ (8A) . Now, we write (B.460) ω(t + is) − β = ω(t) − β + ∂ω(t) · (is) + ∂ω(t) · (−is) + O(s ) where 2 2 2 (B.461) |O(s )|≤D ω 0 × s C (D(0,ρ)) ≤ B × ρ × s −1 ≤ (8A) × s (cf. (2.60)). Note that since ω is real-symmetric, ∂ω(r) and ∂ω(r) are real when r is real. Hence if |ω(t + is) − β| <ν for some t + is ∈ U, one gets by taking the imaginary part in −1 (B.460), using (B.459), ∂ω(t)∈[A , A] and (B.461), that (B.462) |s|≤ (4/3)Aν. This, (B.461) and taking the real part of (B.460) show that (B.463) |ω(t) − β|≤|#(ω (t + is) − β)|+|#(O(s ))|≤ (7/6)ν. −1 Because t → ω(t) is increasing with a derivative bounded below by A (this is the twist condition) the set of t ∈] − ρ, ρ[:= D(0,ρ) ∩ R such that |ω(t) − β|≤ (7/6)ν is a (possibly empty) interval I of length ≤ (7/3)Aν . 2 −1 Lemma B.2. — Let ν ∈]0,(6A B) [.If I is not empty there exists a unique c ∈] − ρ − β β 2Aν, ρ + 2Aν[ such that ω(c ) = β, ω(D(c , 3Aν)) ⊃ D(β, ν). β β Proof. — The uniqueness of c comes from the fact that R t → ω(t) ∈ R is in- creasing. For the existence of c we just notice that if I ⊂] − ρ, ρ[ there is nothing to β β prove (notice that ω is increasing) and otherwise for some ε ∈{−1, 1}|ω(ερ) − β|≤ −1 (7/6)ν.But then,the fact that A ≤ ∂ω(t) ≤ A shows the existence of a unique c ∈[−ρ − (7/6)Aν, ρ + (7/6)Aν] such that ω(c ) = β . β β −1 −1 If we set a = ∂ω(c ) and b = ∂ω(c ) one has a ≥ A , |b|≤ (7A) (cf. (B.459) β β 2 −1 and the fact that ν ∈]0,(6A B) [) and the linear map w → Dω(c )w = aw + bw is invertible, the norm of its inverse being ≤ (7/6)A. Next, we observe that because |ω(c + β 138 RAPHAËL KRIKORIAN w) − ω(c ) − Dω(c )w|≤ B|w| /2and ω(c + w ) − ω(c + w) = Dω(c + w + β β β β β t(w − w))(w − w)dt,the map g −1 −1 g : w → Dω(c ) u − Dω(c ) ω(c + w) − ω(c ) − Dω(c )w β β β b β is (7/6)AB-Lipschitz on {|w|≤ } (some > 0) and sends {|w|≤ } into itself pro- −1 vided (7/6)AB ≤ 1and |u|≤ (6/14)A . In particular if one chooses = 3Aν the 2 −1 map g is (1/2)-contracting (remember ν ∈]0,(6A B) [) and the Contraction Mapping Theoremshows that forany |u|≤ ν ≤ (9/7)ν there exists a unique |w|≤ 3Aν such that ω(c + w) = β + u. We can now prove Proposition 2.7. We ﬁrst observe that the computations done in −1 the proof of Lemma B.2 show by the same token that the map w → Dω(0) (ω (w) − ω(0) − Dω(0)w) is ((7/6)ABρ)-Lipschitz on D(0,ρ) hence contracting from (2.60). This implies that ω is injective when restricted to D(0,ρ) ⊃ U. Assume that (2.61)is satisﬁed for no z ∈ U. Then Lemma B.1 tells us that there exists t + is ∈ Usuch that (B.458) holds and in particular that the interval I of Lemma B.2 is not empty. Applying this latter lemma and using the injectivity of ω when restricted to U we see that ω(U D(c , 3Aν)) ⊂ C D(β, ν) which is the searched for conclusion. Appendix C: Illustration of the screening effect We describe in this section an example where the screening effect mentioned in Sec- tion 3.2 is effective. Consider U as in (3.73) (C.464)U = D(0,ρ) D(c ,ρ ) ⊃ D(0,σ ) j j j=1 (j = 1,..., N, D(c ,ρ ) ⊂ D(0,ρ), ρ σ ), with j j j β γ −N −N ρ = 1,σ = e < 1/10,ρ = ρ := e (j ∈{1,..., N}), j 1 c = (1/4) + (2j − 1)/(4N) ∈[1/4, 3/4]. We assume N 1. The function ϕ(·) := ω (·,∂ D(0,σ )) is harmonic on U D(0,σ ), equal to 1 on ∂ D(0,σ ) and equal UD(0,σ ) to0on ∂ U = ∂ D(0, 1) D(c ,ρ ). Note that the minimum value of the Dirichlet j 1 j=1 integrals |∇ψ(x + iy)| dxdy UD(0,σ ) ψ ∈ H (U D(0, σ )), ψ = 1,ψ = 0 | ∂ D(0,σ ) | ∂ U ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 139 (H denotes the usual Sobolev space) is achieved at ϕ, hence there exists a constant C independent of N such that (C.465) |∇ϕ(x + iy)| dxdy ≤ C; UD(0,σ ) (the constant C is for example the Dirichlet integral of any ﬁxed C function ψ equal to 1on ∂ D(0,σ ) andto0on U D(0, 1/5)). We now use a result by Rauch and Taylor [40] that we adapt to the case of the complex plane: let C be the holed rectangle U ∩ ([1, 4, 3/4]+ −1[−H, H]) with H = C δ ln(δ/ρ ), where δ = 3/(8N), 1 1 C being some large constant. The set C can be covered by the N 1-holed rectangles 1 H U ∩ ([c − δ, c + δ]+ −1[−H, H]), j ∈{1,..., N}, each point of C belonging to no j j H more than two of these holed rectangles. An adaptation of Inequality (4.1) of [40]tothe case of domains in R ( C) asserts that in this situation there exists a constant C > 0 (independent of ϕ, H, N) such that −1 |∇ϕ(x + iy)| dxdy C 2 |ϕ(x + iy)| dxdy H(H + δ ln(δ/ρ ) which in view of (C.465) and the choices made for H and δ implies 2 −(1−γ−) |ϕ(x + iy)| dxdy ≤ CC (C + 1)δ ln(δ/ρ ) N , 2 1 1 because ρ = exp(−N ). In particular, for any μ> 0 there exists a positive constant C 1 μ and a set C of relative measure 1 − μ in C such that for any z ∈ C H,μ H H,μ −(1−γ−μ) |ω (z,∂ D(0,σ ))|=|ϕ(z)|≤ C N , UD(0,σ ) an inequality which is in sharp contrast with (3.71), especially if 1 − γ − μ>β . Getting a useful estimate like (3.72) by this technique is therefore doomed to fail if α< 1− γ − μ. Appendix D: First integrals of integrable ﬂows This section is dedicated to the proof of the following Lemma, on ﬁrst integrals of the integrable ﬂow φ , that was used in the proof of Lemma 5.1. J∇ r Lemma D.1. — Let U be a σ -symmetric open connected set of D and F ∈ O (W ) such σ h,U that (D.466) ∀ t ∈ R, F ◦ φ = F. J∇ r 140 RAPHAËL KRIKORIAN Then, there exists F ∈ O (U) such that on W one has σ h,U F = F ◦ r. Proof. — The Lemma is clear when we are in the (AA) case since the identity ∀ t ∈ R F(θ + t, r) = F(θ , r) clearly implies that F does not depend on θ.Soweconsiderthe (CC)-case. We shallprove that forevery (z,w) ∈ W there exists an open neighborhood V h,U z,w of (z,w) and a holomorphic function f such that F = f ◦ r on V . z,w z,w z,w We consider three cases: 1) If (z,w) = (0, 0) ∈ W . One can write for μ small enough and (z,w) ∈ D(0,μ) , h,U k l F(z,w) = F z w . The identity (D.466) implies M (F) = 0for n = 0hence from k,l n k,l∈N (5.103) one has F(z,w) = M (F)(z,w) = F (zw) and we can choose f (r) = 0 k,k 0,0 k∈N (ir) . k∈N it 2) If zw = 0, with for example w = 0. Then, from (5.106), t → F(0, e w) is holomor- h 1/2 phic with respect to t ∈ R + i]− ln(e ρ /|w|),∞[ and constant on the real axis; it h 1/2 is hence constant on R + i]− ln(e ρ /|w|),∞[. In particular taking t = is, s ∈ R , −s gives F(0, e w) = F(0,w) and by making s →∞ we get F(0,w) = F(0, 0) (notice that (0, 0) ∈ W in that case). The same argument shows that for ( z, w) ∈ W the function h,U h,U −it it h 1/2 h 1/2 t → F(e z, e w) is constant on t ∈ R + i]− ln(e ρ /|w |), ln(e ρ /| z|)[.Now if ( z, w) h 1/2 is close enough to (0,w),inparticularifthere exists 0 < s < ln(e ρ /| z|) such that s −it it s −s 2 |w |/μ < e <μ/| z|, one has with t = is, (e z, e w) = (e z, e w) ∈ D(0,μ) .By(D.466) −it it and point 1), one gets F( z, w) = F(e z, e w) = f (−i zw) . 0,0 3) Otherwise, we can assume that zw = 0. As before, we can argue that the function −it it t → g (t) := F(e z, e w) is constant on the set z,w h 1/2 h 1/2 R + i]− ln(e ρ /|w|), ln(e ρ /|z|)[. −it Any point ( z, w) ∈ W which is close enough to (z,w) is of the form z = e z, w = h,U it e λw, t close to 0 and λ closeto1.Wethushave −it it −1 F( z, w) = F(e z,λe w) = F(z,λw) = F(z, zw z ) = f ( zw) −1 wherewehavedeﬁned f (r) = F(z, irz ). We have thus proven that for each (z,w) ∈ W there exist a neighborhood V h,U z,w and a holomorphic function f such that F = f ◦ r on V .Now if f ◦ r = f ◦ r z,w z,w z,w z,w z ,w on a nonempty open set, the function f and f coincide on a nonempty open set z,w z ,w and thus there exists a holomorphic extension of f of these two functions such z,w,z ,w that f ◦ r = f ◦ r = f ◦ r on V ∩ V . We can now conclude by using the z,w,z ,w z,w z ,w z,w z ,w connectedness of U. ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 141 Appendix E: (Formal) Birkhoff Normal Forms Our aim in this Section is to recall the proof of the existence and uniqueness of the formal BNF, Propositions 6.1, 6.2. This is of course a standard topic but we tried to develop here a framework that is convenient for the proof of Lemma 6.3. We mainly concentrate on the (AA)-case since the formalism in the (CC)-case is very similar to the one developed by Pérez-Marco in [36]. E.1 Formal preliminaries. E.1.1 For mal series. — Let A be a commutative ring and A[[X ,..., X ]] (d ∈ N ) 1 d n n 1 d the ring of formal power series a X , a ∈ A,X = X ··· X (for short X = d n n n∈N d (X ,..., X )). We denote by v(A) = min{|n|= n + ··· + n , a = 0} the valuation 1 d 1 d n n d of an element A = a X and if B = (B ,..., B ) ∈ (A[[X]]) we deﬁne v(B) = n 1 d n∈N min v(B ). l l For any k ∈ N we deﬁne [A] = a X the homogenous part of A of degree k k k |n|=k k ∞ and we set [A] = [A] (resp. [A] = [A] ). ≤k l ≥k l l=0 l=k n n As usual the product of A = a X and B = b X is AB = d n d n n∈N n∈N n n ( a b )X and the derivative ∂ A = n a X with n = n if j = l and d d k l X l n j l j n∈N k+l=n n∈N n = n − 1is n ≥ 1(if n = 0 the derivative of the corresponding monomial is zero). Note l l l that if A ∈ A[[X]],1 ≤ i ≤ j one has (E.467) [A ... A ] = [A ] ···[A ] . 1 j k 1 k j k 1 j k +···+k =k 1 j n d When A = a X ∈ A[[X]] and B ∈ (A[[X]]) , v(B) ≥ 1 one can deﬁne d n n∈N A ◦ B = a B . n∈N If moreover A is endowed with derivations δ : A → A,1 ≤ i ≤ d ,(it means δ (a + i i b) = δ a + δ b, δ (ab) = (δ a)b+ a(δ b))wedeﬁne (cf. Taylor formula) for each a ∈ A and i i i i i B ∈ (A[[X]]) , v(B) ≥ 1 k k (E.468) a ◦ B = (1/k!)(δ a)B ∈ A[[X]]. k∈N k d Similarly, if A = a X , B, C ∈ (A[[X]]) , v(B) ≥ 1, v(C) ≥ 1, we can also deﬁne d k k∈N (E.469)A ◦ (B, C) = (a ◦ B)C . δ n δ n∈N k k k k 42 k 1 d k 1 d We use a multi-index notation, k = (k ,..., k ),B = B ··· B , k!= k !··· k !, δ = δ ··· δ . 1 d 1 d 1 1 d d 142 RAPHAËL KRIKORIAN Lemma E.1. — For k ∈ N , v(A) ≥ 1, v(B) ≥ 1, v(C) ≥ 2, [A ◦ (B, X + C) − A] δ k is a polynomial in the coefﬁcients of [δ A] ,[B] ,[C] for k + k + k ≤ k − 1, |l|≤ k (this k k k 1 2 3 1 2 3 polynomial being with rational coefﬁcients). n n n n n Proof. — Since (a ◦ B)(X + C) − a X = (a ◦ B)((X + C) − X ) + X ((a ◦ n δ n n δ n δ B) − a ) A ◦ (B, X + C) − A = (I) + (II) l l n δ a δ a n n l m n−m l n (I) := B C X ,(II) = B X m l! l! |l|≥0 |l|≥1 |n|≥1 |n|≥1 m≤n, |m|≥1 and one can conclude using (E.467). Assume now that (A,δ) is endowed with a translation by which we mean an action τ of an abelian group (we suppose it is (R ,+))on A that commutes with the derivations δ . E.1.2 Formal diffeomorphisms. — A formal diffeomorphism of A[[X]] is a triple (α, A, B) (we denote it by f )with A, B ∈ (A[[X]]) with v(B) ≥ 2and where v(A) ≥ 1and α,A,B α ∈ R . We can deﬁne the composition of two such objects: f = f ◦ f ⇐⇒ ε,E,D γ,C,D α,A,B ε = α + γ, E = A + (τ C) ◦ (A, X + B), −α δ F = B + (τ D) ◦ (A, X + B) −α δ with v(E) ≥ 1and v(F) ≥ 2. One can check that the usual algebraic rules for composi- tions are satisﬁed and that each such diffeomorphism has an inverse for composition. Remark E.1. — One of the example we have in mind is the following. Take d = ∗ ω d d d ∈ N , A = C (T ) the ring of real analytic functions on T (taking real values on the n ω d real axis) and the ring of formal power series is A[[r]] = { a (θ )r , a ∈ C (T )}, n n n∈N r = (r ,..., r ). The derivations in this case are δ a = ∂ a if a : (θ ,...,θ ) → R is in 1 d i θ 1 d ω d d C (T ), the translation is τ a = a(·− α) (α ∈ R ) and the formal map f can be α α,A,B written under the more suggestive form f (θ , r) = (θ + α + A(θ , r), r + B(θ , r)) (α,A,B) d d as a formal diffeomorphism of T × R . ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 143 E.1.3 Degree. — In case we can assign a degree deg(a) to each element a of the ring A (it satisﬁes by deﬁnition deg(0) =−∞,for all a, b ∈ A,deg(a + b) = max(deg(a), deg(b)) and deg(ab) = deg(a) + deg(b)) we can associate to each weight p : N → N the set n d (E.470) C(p) = a X ∈ A[[X]], ∀ n ∈ N , deg(a ) ≤ p(n) . n n n∈N By extension if B = (B ,..., B ) ∈ (A[[X]]) we say that B is in C(p) if each B ∈ C(p), 1 d l 1 ≤ l ≤ d .If p, q : N → N we deﬁne p ∗ q(n) = max (p(k) + q(l)). (k,l)∈N ,k+l=n In particular if (E.471) p(n) := |n|= n +···+ n , n = (n ,..., n ) ∈ N 1 d 1 d ∗m one has p ∗ p = p and (p − 1) = p − m. We say that the degree deg is compatible with the derivations δ and the translation τ if for any α ∈ R ,1 ≤ i ≤ d ,deg(τ δ a) ≤ deg(a). α i Remark E.2. — The relevant example for our purpose (proof of Lemma 6.3)will ω d be the following. Take d = d , A = C (T )[t] the set of polynomials in t with coefﬁcients ω d (t) n ω d in C (T ),F (θ ) = a (θ ) + ··· + a (θ )t , a ∈ C (T ),0 ≤ j ≤ n, n ∈ N and a = 0. 0 n j n (t) n The derivations δ ,1 ≤ i ≤ d are deﬁned by δ F (θ ) = (∂ a )(θ ) + ··· + (∂ a )(θ )t , i i θ 0 θ n i i (t) n t the translations τ F (θ ) = a (θ − α) + ··· + a (θ − α)t and the degree deg F = n is α 0 n compatible with both of them. The following facts are easily checked. Assume that A is a ring with derivations δ , 1 ≤ i ≤ d and a compatible degree deg and let (p ) be weights. l l∈N (1) If A, B ∈ A[[X]],A ∈ C(p ),B ∈ C(p ) one has AB ∈ C(p ∗ p ). 1 2 1 2 (2) If A ∈ A[[X]],A ∈ C(p ),lim v(A )=∞ one has A ∈ C(max p ). l l l l→∞ l l l l l∈N Let p be a weight such that p ∗ p ≤ p. Using (E.468), points (1)and (2)wehave (3) If a ∈ A,B ∈ A[[X]],B ∈ C(p) then one has a ◦ B − a ∈ C(deg(a) + p). Lemma E.2. — If A ∈ A[[X]], B, C ∈ (A[[X]]) , v(B) ≥ 1, v(C) ≥ 1 with A ∈ C(p− c ), B ∈ C(p− c ), C ∈ C(p− 1), min(c , c ) ≥ 0,then A◦ (B, C)− A◦ C ∈ C(p− c − c ) A B A B δ A B and A ◦ (B, C) ∈ C(p − c ). δ A Proof. — Recall that A ◦ (B, C) = (a ◦ B)C . From point (3) a ◦ B − a ∈ δ n δ n δ n n∈N n n C(deg(a ) + p − c ) and from point (1) (a ◦ B)C − a C ∈ C((deg(a ) + p − c ) ∗ (p − n B n δ n n B ∗|n| n n 1) ) ⊂ C((deg(a ) + p − c ) ∗ (p−|n|)) hence (a ◦ B)C − a C ∈ C(p − c − c ).By n B n δ n A B 144 RAPHAËL KRIKORIAN using point (2)wehave A ◦ (B, C) − A ◦ C ∈ C(p − c − c ). A similar argument shows δ A B that A ◦ C ∈ C(p − c ) whence the conclusion. Before stating the next lemma we introduce the following deﬁnition: we say that a formal diffeomorphism f is in D(p − 1) if A ∈ C(p) ∩ O(r),B ∈ C(p − 1) ∩ O (r). α,A,B Lemma E.3. — One has (1) Let H ∈ C(p − c),c = 0, 1,and f ∈ D(p − 1).Then H ◦ f ∈ C(p − c). α,A,B α,A,B (2) The composition of two formal diffeomorphisms in D(p − 1) is in D(p − 1). (3) The inverse for the composition of a diffeomorphism of D(p − 1) is in D(p − 1). −1 (4) If f = f , then for any k ≥ 1, [A] , [B] , are polynomials in the coefﬁcients of k k α,A,B α,A,B l l 1 2 [τ δ A] ,[τ δ B] ,k , k ≤ k, l , l ≤ k, |m |,|m |≤ k. m α k m α k 1 2 1 2 1 2 1 1 2 2 Proof. — Items 1 and 2 are consequences of Lemma E.2. For point 3 we just have to prove the result when α = 0. Let us denote by U the operator H → H ◦ f .Notethat v((U − id)H) ≥ v(H) + 1 hence the series H := 0,A,B (U − id) Hconverges in A[[r]] and from 1 and 2 one sees that if H ∈ C(p − c), k=0 −1 c = 0,1, thesameistrue for H. To conclude we observe that −(f − id) = (U − 0,A,B −1 id)(f − id) + (f − id) hence 0,A,B 0,A,B −1 k (E.472) −(f − id) = (U − id) (f − id). 0,A,B 0,A,B k=0 Finally point 4 is a consequence of (E.472) and Lemma E.1. E.2 Formal Birkhoff Normal Forms. — From now on we work in the setting of Re- mark E.2. E.2.1 Formal exact symplectic diffeomorphism. — If F ∈ A[[r]],F=α, r+ O (r) with α ∈ R we deﬁne the formal diffeomorphism f (θ , r) = (θ + A(θ , r), r + B(θ , r)) as sug- gested by the implicit relation ϕ = θ + ∂ F(θ , R), r = R + ∂ F(θ , R), f (θ , r) = (ϕ, R) R θ F or more formally A(θ , r) = ∂ F(θ , r + B(θ , r)), 0 = B(θ , r) + ∂ F(θ , r + B(θ , r)) r θ A = ∂ F ◦ (0, B), 0 = B + ∂ F ◦ (0, B)(cf. E.469).) r δ θ δ Note that to prove the existence and uniqueness of the formal BNF of Section E.2.2 it would be enough to work ω d with A = C (T ). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 145 In this situation we use the more intuitive notations R(θ , r) = r + B(θ , r), ϕ(θ, r) − θ = A(θ , r). We shall call such formal diffeomorphisms f formal exact symplectic diffeomorphisms. The set of all such diffeomorphisms is a group under composition. Let us deﬁne E (p − 1) := {f , F=α, r+ F (θ )r ∈ A[[r]], F ∈ C(p − 1)}. F n |n|≥2 The following result is then a consequence of Lemma E.3. Lemma E.4. — The set E (p− 1) is a subset of D(p− 1) stable by composition and inversion. Remark E.3. — In the (CC)-case the relevant choice for A is C[[t]] and the set of formal series is A[[z,w]]. One can extend in this context the notion of σ - m m symmetry. If F = F z w ∈ A[[z,w]] we say it is σ -symmetric (recall d d n,m 2 (n,m)∈N ×N |n|+|m| d d σ (z,w) = (iw, iz))if F = (i) F for all (n, m) ∈ N × N (F is the complex 2 n,m m,n n,m conjugate of F and i = −1). Similarly one can deﬁne the notion of σ -symmetric n,m 2 formal diffeomorphism. If F is σ -symmetric then f is σ -symmetric. 2 F 2 E.2.2 Existence and uniqueness of the formal BNF. — We prove at the end of this sub- section that given f we can ﬁnd B(r) = 2πω , r+ O (r) ∈ R[[r]] and a formal 2πω ,r+F 0 2 2 exact symplectic diffeomorphism f = id + O (r),Z = O (r) which is normalized in the sense that (E.473) Z(ϕ, Q)dϕ = 0, such that (E.474) f ◦ f (θ , r) = f ◦ f (θ , r). Z 2πω ,r+F B Z Moreover, Z and B are uniquely determined by F. We use the notations f : (θ , r) → (ϕ, R), f : (ϕ, R) → (ψ, Q) so that 2πω ,r+F Z ψ = ϕ + ∂ Z(ϕ, Q), R = Q + ∂ Z(ϕ, Q) Q ϕ ϕ = θ + 2πω + ∂ F(θ , R), r = R + ∂ F(θ , R). 0 R θ Using the relation R = Q + ∂ Z(θ + 2πω + ∂ F(θ , R), Q) and the fact that ϕ 0 R g : (θ , Q, R) → (θ , Q, R − Q − ∂ Z(θ + 2πω + ∂ F(θ , R), Q)) is a formal diffeomor- ϕ 0 R 44 2 −1 phism we can deﬁne R(θ , Q) = Q + O (Q) by (θ , Q, R(θ , Q)) = g (θ , Q, 0). Deﬁned on A[[Q, R]]. 146 RAPHAËL KRIKORIAN Lemma E.5. (1) For any k ≥ 1,the coefﬁcientsof [R(θ , Q)] are polynomials in the coefﬁcients of [τ δ F] , k 2π m ω k 1 0 1 [τ δ Z] ,k , k , l , l ,|m |,|m |≤ k. 2π m ω k 1 2 1 2 1 2 2 0 2 (2) If F, Z ∈ C(p − 1), the formal diffeomorphism (θ , Q) → (θ , R(θ , Q)) is in D(p − 1). Proof. — These are consequences of Lemma E.3. Let f (θ , r) = (θ , r ); from the formal conjugation relation (E.474)weget f ◦ Z B f (θ , r) = (θ +∇ B(r ), r ) = (ψ, Q) hence Q = r and θ = θ + ∂ Z(θ , Q).Wethus Z Q have θ + ∂ Z(θ , Q)+∇ B(Q) = ϕ + ∂ Z(ϕ, Q) Q Q and using the relations between ϕ, θ yields −∂ F(θ , R) = ∂ Z θ + 2πω + ∂ F(θ , R), Q − ∂ Z(θ , Q) R Q 0 R Q − (∇ B(Q) − 2πω ) that we can write (E.475) −∂ F (F, Z) = ∂ Z(θ + 2πω , Q) − ∂ Z(θ , Q) − (∂ B(Q) − 2πω ) Q Q 0 Q Q 0 where F (F, Z) = O (r) is uniquely deﬁned (note that the RHS of (E.475)is O(r))by (E.476) ∂ F (F, Z) = ∂ F(θ , R(θ , Q)) Q Q + ∂ Z θ + 2πω + ∂ F(θ , R(θ , Q)), Q − ∂ Z(θ + 2πω , Q) . Q 0 Q Q 0 We thus have (E.477) −F (F, Z) = Z(θ + 2πω , Q) − Z(θ , Q) − (B(Q) − 2πω , Q). 0 0 Lemma E.6. (1) For any k ≥ 1,the coefﬁcientsof [F (F, Z) − F] are polynomials in the coefﬁcients of l l 1 2 [τ δ F] ,[τ δ Z] ,k , k ≤ k − 1,l , l ≤ k, |m |,|m |≤ k. 2π m ω k 2π m ω k 1 2 1 2 1 2 1 0 1 2 0 2 (2) If F, Z ∈ C(p − 1), one has F (F, Z) ∈ C(p − 1). Proof. — This is a consequence of (E.476), Lemma E.5 and Lemma E.3. From (E.477) one thus has k = 2, −[F] (θ , Q)=[Z] (θ + 2πω , Q)−[Z] (θ , Q)−[B] (Q), 2 2 0 2 2 (E.478) ∀ k ≥ 3 −[F (F, Z)] (θ , Q)=[Z] (θ + 2πω , Q)−[Z] (θ , Q)−[B] (Q). k k 0 k k ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 147 Before completing the proof of the existence and uniqueness of the Birkhoff Nor- mal Form (E.474) we state the following result (the ﬁrst part of which at least is classical; see for example [13]): Lemma E.7. — If ω is Diophantine, for any G ∈ A[[r]] there exists a unique pair (Z, B) with Z ∈ A[[r]] normalized in the sense of (E.473)and B = B(r) ∈ R[t][[r]] such that (E.479)G(θ , Q) = Z(θ + 2πω , Q) − Z(θ , Q) + B(Q). Furthermore: (1) B(Q) = G(θ , Q)dθ and if G =[G] one has Z =[Z] , B =[B] and d k k k the coefﬁcients of [Z] are R-linear functions of the coefﬁcients of [G] ;(2) if G ∈ C(p − 1) then k k Z, B ∈ C(p − 1). −d −il,θ Proof. — If we denote by G(l, Q) = (2π) G(θ , Q)e dθ the l -th Fourier coefﬁcient of θ → G(θ , Q) (l ∈ Z )then(E.479) follows if B(Q) = G(θ , Q)dθ and if Z(θ , Q) is deﬁned by 2π il,ω −1 il,θ Z(θ , Q) = (e − 1) G(l, Q)e . l∈Z {0} The conclusions of the Lemma are clear from the preceding expression. Proof of the existence and uniqueness of the BNF (E.474). – Uniqueness: Equation (E.478), Lemma E.6 and Lemma E.7 show inductively that [Z] ,[B] are uniquely deﬁned by [F] ,2 ≤ j ≤ k − 1. Hence, Z and B are unique. k k j –Existence:Deﬁne [Z] ,[B] by (E.478) and then inductively for k ≥ 3, [Z] ,[B] ,by 2 2 k k (E.480) −[F (F,[Z] )] (θ , Q)=[Z] (θ + 2πω , Q)−[Z] (θ , Q)−[B] (Q) ≤k−1 k k 0 k k k−1 ∞ ∞ where Z = [Z] . Setting F = [Z] ,B = [B] one can check that (E.477) ≤k−1 l l l l=2 l=2 l=2 holds modulo O (r) for any k and hence in A[[r]]. (t) E.3 Proof of Lemma 6.3.— We deﬁne F (θ , r) = tF(θ , r) + (1 − t)G(θ , r) which ω d (t) is in A[[r]] ∩ C(p − 1), A = C (T )[t]. Note that for any k ≥ 2, [F ] ∈ C(p − 1).In particular, as a consequence of Lemmata E.6,item 2 and E.7, point (2), the sequences (t) (t) (t) [Z ] ,[B ] , inductively constructed in (E.480), are in C(p − 1).Hence B (r) := k k b (t)r is in C(p− 1) which by deﬁnition (cf. (E.470), (E.471)) means that the degree d n n∈N in t of each b (t) is ≤|n|− 1. Appendix F: Approximate Birkhoff Normal Forms We give in this section the proofs of Propositions 6.4 and 6.5. 148 RAPHAËL KRIKORIAN F.1 A useful lemma. — Let be given for each α ∈]0, 1/2[,afunction P : R × α + R → R ,P : (k, t) → P (k, t) nondecreasing in each variable and assume that + + α α (ε ) ,I ⊂ N is a sequence of nonnegative real numbers depending on α ∈]0, 1/2[ α,k k∈I α and deﬁned inductively as long as a condition of the form (F.481)P (k,ε )< 1 α α,k ∗ ∗ is satisﬁed (we assume that ε satisﬁes (F.481)). Let us call J = 0, k , k ≥ 1the maxi- α,0 α α α mal set of integers k ∈ N for which ε is deﬁned: this means that if k ∈ J and ε satisﬁes α,k α α,k ∗ ∗ (F.481)then k + 1 ∈ J (in particular P (k ,ε ) ≥ 1). Let θ> 0, a > 0and k ∈ N be α α α,k θ,a α α such that ∗ θ −a (F.482) ∀k ∈ 0, min(k , k ) − 1,ε ≤ C α × 1 + α ε ε . θ,a α,k+1 θ,a α,j α,k j=0 We have the following type of Gronwall Lemma: Lemma F.1. — Assume that θ a (F.483) (2C α) < 1/2,ε ≤ α /2, P (k + 1,ε )< 1. θ,a α,0 α θ,a α,0 Then, (F.484) k ≥ k θ,a and θ k (F.485) ∀ k ∈[0, k ]∩ N,ε ≤ (2C α) ε . θ,a α,k θ,a α,0 ∗ ∗ Proof. — 1) Let k = min(k , k ). We ﬁrst prove that the set θ,a α,θ ,a α ∗ θ k K ={k ∈ 0, k ,ε >(2C α) ε } α,θ ,a α,k θ,a α,0 α,θ ,a is empty. If this were not the case we could deﬁne k = inf K and write α,θ ,a α,θ ,a θ k (F.486) ∀ k ∈ 0, k − 1,ε ≤ (2C α) ε α,θ ,a α,k θ,a α,0 hence α,0 ε ≤ ≤ 2ε α,j α,0 1 − (2C α) α,θ ,a j=0 and thus from (F.482)and (F.483), for all 0 ≤ k ≤ k − 1, α,θ ,a θ −a θ ε ≤ C × α (1 + 2α ε )ε ≤ (2C α )ε . α,k+1 θ,a α,0 a,k θ,a α,k ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 149 This implies that for all 0 ≤ k ≤ k one has α,θ ,a θ k ε ≤ (2C α) ε α,k θ,a α,0 θ k α,θ ,a and in particular ε ≤ (2C α) ε . This contradicts the deﬁnition of k as α,k θ,a α,0 α,θ ,a α,θ ,a inf K . α,θ ,a 2) Since K is empty, one has α,θ ,a ∗ kθ (F.487) ∀ k ∈ 0, k ,ε ≤ (2C α) ε ≤ ε . α,k θ,a α,0 α,0 α,θ ,a But since k + 1 ≤ k + 1and (2C α) ≤ 1 θ,a θ,a α,θ ,a P (k + 1,ε ) ≤ P (k + 1,ε ) α α,k α θ,a a,0 α,θ ,a α,θ ,a < 1 ∗ ∗ ∗ ∗ This implies that k + 1 ∈ J hence k < k and from the deﬁnition of k ,weget α,θ ,a α,θ ,a α α,θ ,a ∗ ∗ k ≥ k which in its turn implies k = k .Wehavethusproventhatfor all k ≤ k θ,a θ,a θ,a α α,θ ,a one has kθ ε ≤ (2C α) ε . α,k θ,a α,0 F.2 Proof of Proposition 6.4 (BNF, (CC)-Case). — For n large enough we deﬁne (F.488) ρ = , W = e W 0 0 h,D(0,ρ ) and for k ≥ 0 we introduce the sequences −1 (F.489) δ = C , δ ≤ h/10 k l (k + 1)(ln(k + 2)) l=0 k−1 k−1 (F.490) ρ = exp(− δ )ρ , W = exp(− δ )W . k l 0 k l 0 l=0 l=0 Recall that a = max(2a + 1, 30) and that for some m ≥ a (cf. (6.141)) 1 1 2m m (F.491)F (z,w) = O (z,w), ε := F ρ . 0 0 0 W The choice of these sequences, in particular the choice of a summable sequence δ , is not necessary (we only perform a ﬁnite number of conjugation steps) and is indeed not the best one insofar as it leads to a(n) (arbitrary small) loss β in the exponent. The “optimal” one is (after loosing a ﬁxed fraction of h in the ﬁrst step) to choose uniformly at each ∗ ∗ step δ ∼ h/k so that δ ∼ h where k is the ﬁnite number of conjugation steps we perform. k ∗ k 0≤k≤k 150 RAPHAËL KRIKORIAN We shall construct inductively for k ≥ 0, sequences Z ∈ O (W ),F ∈ O (W ) ∩ k σ k k σ k k+2m 2 O (z,w), ∈ O (D(0,ρ )), (r) = 2πω r + O (r),suchthat k σ k k 0 = , F = F 0 0 and for k ≥ 1, −1 (F.492) g ◦ ◦ f ◦ g = ◦ f . F k F k 0 0 k k −1 To do this we proceed the following way: assuming (F.492) holds and δ ≤ q ,weap- ply Proposition 5.3 with τ = 0, K/2 = N = q cf. (6.143), and deﬁne Y ∈ O (W ) ∩ n k σ k k+2m O (z,w) (see Remark 5.3) satisfying (F.493) −[Q ]· Y = T F − M (F ), Y −δ /2 q F 0 k q k 0 k k k k W n e W k k n where we denote (F.494)Q (r) = 2πω r. 0 0 Using Lemma 5.4 we get (cf. formula (5.124)) −1 f ◦ ◦ f ◦ f = ◦ f (a) Y F +M(F ) ˙ k k k Y k k k F −M (F )+[ +M(F )]·Y +O (Y ,F ) k 0 k k k k k k = ◦ f (a) . +M(F ) ˙ k k R F +[ +M(F )]·Y −[Q ]·Y +O (Y ,F ) q k k k k 0 k k k n 2 −1 Hence f ◦ ◦ f ◦ f = ◦ f with Y F F k k k Y k+1 k+1 (F.495) − = M(F ) k+1 k k and, using the fact that [ + M(F )]· Y −[Q ]· Y = O(|∇ Y |×|∂( − Q ) ◦ r|) + k k k 0 k k k 0 (2) O (Y , F ), k k (a) (F.496)F = R F + O (Y , F ) + O |∇ Y |×|∂( − Q ) ◦ r| . k+1 q k k k k k 0 n 2 j+2m Notice that from (F.495) and the fact that for 0 ≤ j ≤ k − 1, M (F ) = O (z,w) (cf. the 0 j remark at the end of Section 5.1.1), hence M(F)(r) = O (r); one thus has (F.497) ∀ 0 ≤ j ≤ k, (r) − Q (r) = O(r ). k 0 (a) k+2m 2k+4m−2a Since F , Y ∈ O (z,w) we have (see Remarks 2.1, 5.3) O (Y , F ) = O (z,w); k k k k k+2m+1 q also, O(|∇ Y|×|∂( − Q ) ◦ r|) = O (z,w) and from (5.115)R F = O (z,w) k 0 q k (if q ≥ m). As a consequence, since 2k + 4m − 2a ≥ k + 1 + 2m (m ≥ 2a + 1) we see that n ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 151 k+1+2m 2 F = O (z,w). Furthermore since (r) − Q (r) = O(r ) k+1 0 0 ∂( − Q ) ◦ r ≤∂( − ) ◦ r +∂( − Q ) ◦ r k 0 W k 0 W 0 0 W k k k ∂( − ) ◦ r + sup|r(W )| k 0 W k ∂( − ) ◦ r + ρ k 0 W k and using (2.55) −1/2 −1 ∇ Y −δ δ ρ Y −δ /2 k k k k e W k e W k k k −1/2 2 −1 q ρ δ F k W k k n k hence 1/2 2 −2 −2 2 −1 (F.498) |∇ Y |×|∂( − Q | q ρ δ − F + q δ ρ F . k k 0 k 0 D(0,ρ ) k W k W k k k k n k k n k From (F.496), (F.493), (F.498), and (5.112) we get that, provided −a −a 2 ρ δ q F < 1, k W k k n one has the inequalities −1 −q δ /2 −a −a 2 2 n k −δ (F.499) F δ e F + ρ δ q F k+1 k k W k e W k k k k k n W 1/2 2 −2 −2 2 −1 + q ρ δ − F + q δ ρ F k 0 D(0,ρ ) k W k W n k k k k n k k k and (F.500) − −δ F . k+1 0 D(0,e ρ ) j W j=1 Let us deﬁne s = − ,ε =F ; k k 0 D(0,ρ ) k k W k k then, one has 1/2 ⎨ −1 −q δ /2 −a 2 2 −2 2 −1 ε δ e + (ρ δ ) q ε + q (ρ δ ) s + q δ ρ ε k+1 k k k k k k k k k k n n n (F.501) s ε k+1 j j=0 as long as −a −a 2 ρ δ q ε < 1. k k n 152 RAPHAËL KRIKORIAN Let k be the largest integer for which the preceding sequences are deﬁned and satisfy the stronger condition −1/5 ∗ −a −a 2 (F.502) ∀ k < k , P (k,ε ) := ρ ρ δ q ε < 1. q k k n 0 k k n −(1+σ) From (F.489)for any σ> 0 one has δ (k + 1) .For θ ∈]0, 1/6[ deﬁne k σ 1 1 μ = − θ, μ = μ . 0 0 6 1 + σ Since ρ ρ one has k 0 −1/6+μ 0 −θ q δ ρ = ρ n k ⎨ 0 0 ∗ −μ −1−μ −7/6+θ −1 0 ∀ k < min(k ,ρ ), (ρ δ ) ρ = ρ k k 0 0 −1 1/2 −1/3+1/2−μ 2 0 θ q δ ρ ρ = ρ , k k 0 n 0 −μ 1/5 −1 −q δ /2 n k thus (note that for k <ρ , ρ 1), δ e ≤ ρ and consequently, if ρ 1, 0 θ 0 θ 0 k 0 1/5 1/5 −1/3−7/3+2θ ∗ −μ θ ∀ k < min(k ,ρ ), ε (ρ + ρ + ρ ε + ρ )ε k+1 j k 0 0 0 0 0 j=0 θ −3 ≤ C ρ (1 + ρ ε )ε . θ j k 0 0 j=0 1/5+(7/6)a+1/3 2a+1 Since from (F.491) ε ≤ ρ <ρ we see that condition (F.483) of Lemma F.1 0 0 −μ is satisﬁed (with α = ρ , k = ρ )hence 0 θ,α −μ θ k (F.503) ∀ k ∈[0,ρ ]∩ N,ε ≤ (2C ρ ) ε . k θ 0 0 −μ 1−β Now for any 0 <β 1, one can choose θ and σ so that ρ = q and in particular 0 n 1−β taking k = k =[q ] and using (F.501)one gets for q 1 β n β 1−β −q (F.504) ε ≤ e , s ≤ 2ε k k 0 (we can assume a > 10). We now deﬁne BNF BNF F = F, = , −1 −1 k k q q n n and BNF −1 −1 g = f ◦···◦ f −1 Wh Wh q Y Y k ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 153 Wh 2 −δ /2 where Y is a C Whitney extension of (Y , e W ) given by Lemma 2.2; one has (cf. j j (F.493), (F.491)) BNF 2 θ k −4 −4 8 −(m−26) (F.505) g − id 1 q (2C ρ ) ε ρ h k q . −1 C θ 0 0 θ,h n 0 n k=0 k+2m Inequalities (F.504)and (F.505), and the fact that F ∈ O (r), give the conclusion 1−β BNF q of Proposition 6.4.Notethat(6.145) is a consequence of F ∈ O (z,w) and −1 Remark 6.2. For the last statement of the Proposition, we can choose = + k−1 Wh Wh 3 M(F ) where F is an C Whitney extension of (F , W ) given by Lemma 2.2. j j j j=0 J F.3 Proof of Proposition 6.5 (BNF (AA) or (CC) Case, ω Diophantine). — The proof, that we mainly illustrate in the (AA)-Case, as well as the notations, are essentially the same as the ones of the proof of Proposition 6.4 (see Section F.2, in particular, we use the deﬁnitions (F.489), (F.490)for δ , ρ ,W ) with the following differences: k k k – we replace (F.488)by (τ+1)/γ ρ = ρ , W = W = W bτ 0 0 h,D(0,ρ ) h,D(0,ρ ) where γ = 1 in the (AA)-case and γ = 1/2 in the (CC)-case. – since ω is Diophantine, we can and do solve instead of (F.493) the equation without −1 truncation (using Proposition 5.3 with N=∞,K = κ , cf. (6.149)): −τ −[Q ]· Y = F − M (F ), Y −δ /2 δ F 0 k k 0 k k e k W k W k k where Q (r) = 2πω r,cf. (F.494). 0 0 Notice that both in the (AA) and (CC) cases m m (F.506)F = O (r), F ≤ ρ . 0 0 W 0 0 In place of (F.496) we get (in the (AA)-case) (a) (F.507)F = O (Y , F ) + O |∂ Y |×|∂( − Q ) ◦ r| . k+1 k k θ k k 0 k+m Since F , Y ∈ O (r)(θ ) (see Remarks 2.1, 5.3)wehave O(|∂ Y |×|∂( − Q )◦ r|) = k k θ k k 0 (a) k+m+1 2k+2m−a O (r)(θ ) and O (Y , F ) = O (r)(θ ) (a from Lemma 5.4). As a consequence, k k k+1+m since 2m ≥ a we see that F = O (r)(θ ). k+1 From (F.507) and the fact that (cf. (2.54)) −1 ∂ Y −δ δ Y −δ /2 , ∂( − Q ) ◦ r ≤ ρ θ k k k k 0 0 W k e W k e W k k k 154 RAPHAËL KRIKORIAN hence −(1+τ) |∂ Y |×|∂( − Q ) ◦ r| δ ρ F θ k 0 0 W k k k where γ = 1 in the (AA)-case. A similar computation (cf. (F.498)) shows that one can take γ = 1/2 in the (CC)-case. With the notations s = − , ε =F ,wethen k k 0 D(0,ρ ) k k W k k get −(1+τ) γ −(τ+a) −(2+τ) ε (ρ δ ) ε + (ρ δ ) s + δ ρ ε k+1 k k k k k k k k k (F.508) s ε k+1 j j=0 provided −(τ+a) (F.509) (ρ δ ) ε < 1. k k k Let k be the largest integer for which the preceding sequences are deﬁned and satisﬁes −γ −(τ+a) (F.510) ∀k < k , P (k,ε ) := ρ (ρ δ ) ε < 1; ∗ ρ k k k k 0 0 the condition involved in (F.510) implies (F.509). From (F.489)for any σ> 0 one has −(1+σ) δ (k + 1) .Fix θ ∈]0,γ[ and deﬁne k σ γ − θ 1 μ = . 1 + τ 1 + σ Since ρ ρ one has k 0 −1−μ(1+σ) −1 (ρ δ ) ρ −μ k k ∗ 0 ∀ k < min(k ,ρ ), 0 −(1+τ) γ γ−(1+τ)μ(1+σ) δ ρ ρ = ρ . k k 0 0 If we set γ − θ a = 1 + (2 + τ) + θ ≤ (3/2)(2 + τ) + 1 1 + τ we then get using (F.510)and (F.508) −μ γ ∗ −a+θ θ ∀ k < min(k ,ρ ), ε ≤ C ρ + ρ ε + ρ ε k+1 σ j k 0 0 0 0 j=0 θ −a ≤ C ρ × 1 + ρ ε ε . σ j k 0 0 j=0 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 155 We now apply Lemma F.1 with α = ρ : since (a > 10) γ − θ max a,(1 + )(τ + a) + 1 ≤ 2(τ + a) ≤ a ≤ m; 1,τ 1 + τ −μ condition (F.506)shows that (F.510) is satisﬁed with k = min(k ,[ρ ]) as well as condi- −μ tions (F.483)for ρ 1. We thus get if k := [ρ ], 0 θ,σ θ k (F.511) ∀ k ∈[0, k]∩ N,ε ≤ ρ ε . k 0 Since θ and σ can be taken arbitrarily close to 1, for any 0 <β 1 one has for ρ 1 −(1−β) −ρ ε ≤ e , s ≤ 2ε . k k We conclude like in the proof of Proposition 6.4 (Section F.2) by deﬁning BNF BNF BNF −1 −1 F = F, = , g = f ◦···◦ f . k k Wh Wh ρ ρ ρ Y Y 1−β k+a BNF (1/ρ) 1,τ Note that since F ∈ O (r) one has F ∈ O (r) and (6.154) is a consequence of Remark 6.4. Appendix G: Resonant Normal Forms In this section we shall only consider the (AA)-Case. Let c ∈ D(0, 1), ρ> 0, h > 0and h h (G.512) ∈ O (e D(c, ρ)) and F ∈ O (e W ) σ σ h,D(c,ρ) where satisﬁes the twist condition (A, B ≥ 1) −1 −1 2 −1 3 ∀ r ∈ R, A ≤ (2π) ∂ (r) ≤ Aand (2π) D ≤ B. Our aim in this section is to give an approximate Normal Form for ◦ f in a neighbor- hood of a q-resonant circle by which we mean that for some (p, q) ∈ Z × N , p ∧ q = 1 −1 (2π) ∂(c) = . This Normal Form is quite similar in spirit (and in its construction) to the approximate BNF. It is used in the paper in Sections 8 (approximate Hamilton-Jacobi Normal Form) and 15 (creating hyperbolic periodic points). As usual we deﬁne ω(c) := ∂(c). 2π 156 RAPHAËL KRIKORIAN Proposition G.1 (q-resonant Normal Form).— There exists a universal constant a ≥ 10 (not depending on q and that we can assume in N) such that, if one has −8 ρ< (Aq) (G.513) F < ρ , h,D(c,ρ) −1/q then the following holds: There exist ∈ O (D(c, e ρ)) ∩ TC(2A, 2B), res cor −1/q −1/q F , F ∈ O (e W ), g ∈ Symp (e W ) σ h,D(c,ρ) RNF h,D(c,ρ) ex,σ such that −1 res cor g ◦ ◦ f ◦ g = ◦ ◦ f ◦ f F RNF 2π(p/q)r F RNF (G.514) res res F is 2π/q − periodic, M (F ) = 0, where ⎪ −1/q − ( − 2π(p/q)r) F D(c,e ρ) W ⎨ h,D(c,ρ) res (G.515) −1/q F F e W h,D(c,ρ) h,D(c,ρ) −2 a −5 g − id 1 (qρ ) F ≤ ρ RNF C h,D(c,ρ) and cor −1/4 (G.516) F −1/q exp(−ρ )F . e W W h,D(c,ρ) h,D(c,ρ) We give the proof of this Proposition in the next subsections. Remark G.1. — The implicit constants in the symbol of the preceding estimates depend on h;if h > 0, they can be bounded above by a constant C whenever h ≥ h . 0 h 0 G.1 A preliminary Lemma. Lemma G.2. — 1) For any (k, l) ∈ Z × Z one has ⎨ either q|kand p|l (G.517) p 1 ⎩ or |k × − l|≥ . q q 2) Let −1 N = (qρA) . For any r ∈ D(c, ρ) and any (k, l) ∈ N × Z, 1 ≤ k ≤ N, which is not in (q, p)Z one has (G.518) |kω(r) − l|≥ 1/(2q). ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 157 Proof. — 1) Indeed |k(p/q) − l|=|kp − lq|/q and if the integer kp − lq is 0 then q|k and p|l . 2) We just notice that |kω(r) − l|≥|kω(c) − l|− k|ω(r) − ω(c)| ≥ (1/q) − N∂ω ρ D(c,ρ) ≥ 1/(2q). Deﬁne (see Section 5.1) q−res q−nr q−res (G.519)T F = M (F), T F = T F − T F. k N N N N k∈Z |k|<N q|k q−res q−nr res nr We shall often use in what follows the shortcuts T and T for T ,T . N N N N From (5.101), (5.111) we see that res 2π/q res (G.520)T F ◦ φ = T F N r N and res −1 (G.521) T F −δ δ F . e h,D(c,ρ) h,D(c,ρ) Corollary G.3. — For any F ∈ O (W ), there exists Y ∈ O (W ) such that σ h,D(c,ρ) σ h,D(c,ρ) M (Y) = 0 and nr (G.522)T F=[]· Y. This Y satisﬁes for any 0 <δ < h −1 (G.523) Y −δ qδ F . e W W h,D(c,ρ) h,D(c,ρ) Proof. — This is a simple adaptation of the proof of Proposition 5.3 (the non- resonance condition is replaced by (G.518)). G.2 Elimination of non-resonant terms. Proposition G.4. — There exists a universal constant a (not depending on q) such that if −1 N = (qρA) and 1/8 −1 ρ <(qA) (G.524) F < ρ , h,D(c,ρ) 158 RAPHAËL KRIKORIAN res nr −1/q −1/q then there exist F , F ∈ O (e W ),g ∈ Symp (e W ) such that σ h,D(0,ρ) h,D(0,ρ) ex,σ −1/q −1 (G.525) [e W ] g ◦ ◦ f ◦ g = ◦ f res nr h,D(0,ρ) F F +F res res (G.526)F = T (F + O(qρF )) N h,D(c,ρ) res F being 2π/q-periodic and −2 2 a −5 (G.527) g − id 1 (qρ ) F ≤ ρ C h,D(c,ρ) nr −1/3 (G.528) F −1/q q exp(−ρ )F . e W h,D(c,ρ) h,D(c,ρ) Proof. — Note that we can assume, using Lemma 2.2 that F ∈ O (W ) and σ h,D(c,ρ) satisﬁes a a −7 3 3 −1/(10q) 3 (G.529) ε := F ≤ ρ , F ρ 0 C e W h,D(c,ρ) where a will be deﬁned in (G.539). −1 −1/(10q) Let N = (qρA) .Wedeﬁne W = e W , ρ = ρ and for k ≥ 1 0 h,D(c,ρ ) 0 k−1 k−1 −1 (5q) (G.530) δ = ,ρ = exp(− δ )ρ, W = exp(− δ )W k k l k l 0 4/3 (k + 1) l=0 l=0 nr res nr res and we construct sequences Y , F , F , F ∈ O (W ) such that F = F + F k k σ k k k k k k res res nr res res nr (G.531)F = T F, F = F − F , T F = 0 0 N 0 0 N 0 and for k ≥ 0 −1 nr res nr res (G.532) f ◦ ◦ f ◦ f = ◦ f Y F +F F +F k Y k k k k+1 k+1 where for any k, 2π/q res res F ◦ φ = F . k J∇ r k −δ /2 By Corollary G.3 there exists Y ∈ O (e W ) such that k σ nr nr −1 nr (G.533) []· Y =−T F , Y −δ /2 qδ F . k k k W N k e W k k k res nr Let F := F + F and compute using Proposition 4.7 k k −1 nr res nr res f ◦ ◦ f ◦ f = ◦ f Y F +F F +F +[]·Y +DF O (Y ) −δ /2 1 k Y k k k k k k k k k e W −1 = ◦ f nr res nr res nr R F +T F +F +qδ DF O (F ) N k −δ /2 1 k N k k k k e k W nr res = ◦ f F +F k+1 k+1 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 159 with nr nr −2 nr F = R F + q(ρ δ ) F O (F ) N k k k W 1 k+1 k k k (G.534) res res nr res F = T F + F . k+1 N k k nr res nr res nr res In particular since F = F + F = F + T F + F k+1 k+1 k+1 k+1 N k k nr res nr nr (G.535)F = F + F + T F − F . k+1 k k+1 N k k ∗ ∗ If we deﬁne ε =F , ∗= nr, res, ε =F we get from (G.531)and W k k W k k k k Lemma 5.2 nr −1 (G.536) ε δ ε qε 0 0 0 0 and from (G.534), (G.535) and Lemma 5.2 that for some a > 0 nr −1 −δ N/2 nr −a nr ε δ e ε + qε (ρ δ ) ε k k k ⎨ k+1 k k k res res −1 nr (G.537) ε ε + δ ε k+1 k k −1 nr nr ε = ε + O(ε + δ ε ) k+1 k k+1 k k provided for some a > 0 (that we can assume ≥ 4) −a nr (G.538) (ρ δ ) ε < 1. k k From now on we deﬁne a = 2a + 2, see (G.529), (G.539) ε ≤ ρ with a = 2a + 2 ≥ 10; 0 3 notice that this implies (see (G.536)) nr 2a (G.540) ε ≤ ρ . 0 0 ∗ nr res Let k be the largest integer for which the sequences ε ,ε ,ε are deﬁned. We notice k k −1/3 that for k < min(k ,ρ ) one has from (G.524) −4/9 −1/8 −4/9 −1 −1 −1 −1 −2 (ρ δ ) ρ qρ A ρ ρ ρ ≤ ρ . k k 0 0 0 0 0 0 k −1/3 −4/9 k −1 nr nr Since ε = ε + O(δ ε ) we get that for k+ 1 ≤ ρ , ε = ε + O(ρ ε ) k 0 k 0 k j=0 k 0 0 j=0 k −1/3 −1/8 hence if k < min(k ,ρ ) (recall q ≤ ρ ) 0 0 −a nr −(2a+1) −(2a+2) nr nr qε (ρ δ ) ε ≤ C ρ ε + ρ ε ε . k k k 0 k 0 0 k k j=0 −3/4 −1 −1 −2 −1 −1 On the other hand, from (G.524), q N = q ρ A ≥ A ρ hence, if k + 1 ≤ 0 0 −1/3 −3/4 4/9 −11/36 −1 −1 4/3 −δ N ρ one has δ N = q N/(k + 1) ≥ ρ ρ = ρ and thus δ e ≤ ρ if ρ k 0 0 0 0 0 0 k 160 RAPHAËL KRIKORIAN −1/3 is small enough. The outcome of this is that for k + 1 ≤ ρ one has (we use condition (G.539)) nr −(2a+3) nr nr ε ≤ Cρ 1 + ρ ε ε . k+1 0 k k j=0 Since for ρ 1 one has (cf. (G.540)) nr 2a a (G.541) ε ≤ ρ ≤ (ρ δ ) 0 0 0 0 and we can thus apply Lemma F.1 with α = ρ ,toget −1/3 ∗ nr k nr −k (G.542) k ≥ ρ , ∀0 ≤ k ≤ k ,ε ≤ (2Cρ ) ε ≤ e qε . ∗ 0 0 0 k 0 We now set res res nr nr −1 Wh F = F , F = F , g = f ◦···◦ f ∗ ∗ Wh k k Y ∗ k −1 Wh 2 −δ /2 where Y is a C Whitney extension of (Y , e W ) given by Lemma 2.2. The con- j j jugation relation (G.525) then holds and the conclusion (G.528) is satisﬁed since from (G.542) −1/3 −2/q nr −ρ e W ⊂ W ∗ , F −2/q e qε . h,D(c,ρ) k e W 0 h,D(c,ρ) To check (G.527) we just notice that from (G.533) 2 −4 g − id 1 q ρ ε . C h 0 res res res ∗ nr res nr Finally, since F = T F + T ( F ) and T F = 0(cf. (G.531)) one has from the N N k=0 k N 0 nr k nr −(k−1) inequality ε ≤ (2Cρ ) ε ρ e ε (ρ 1, k ≥ 1) 0 0 0 0 k 0 k k k ∗ ∗ res res res nr nr nr F = T F + T ( F ), F ≤ ε qρ ε W ∗ 0 0 N N k k k k k=1 k=1 k=1 which gives conclusion (G.526): res res F = T F + O(qρε ) . G.3 Proof of Proposition G.1.— We apply Proposition G.4 and we write using Lemma 4.6 ◦ f nr res = ◦ ◦ f res ◦ f nr res nr F +F 2π(p/q)(r−c) −2π(p/q)(r−c) F F +DF O (F ) W 1 h,U = ◦ ◦ f res ◦ f cor 2π(p/q)(r−c) −2π(p/q)(r−c) F F ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 161 with cor a −a nr (G.543) F −1/q q ρ F −2/q e W e W h,D(c,ρ) h,D(c,ρ) provided for some a > 0 a −a nr q ρ F −2/q < 1. e W h,D(0,ρ) The inequalities (G.524)and (G.528) show that this last condition is satisﬁed if ρ 1. We now observe that res res f = ◦ f F M (F) F −M (F) 0 0 and that ◦ = . −2π(p/q)(r−c) M (F) −2π(p/q)(r−c)+M (F) 0 0 If we set g = g RNF res res = − 2π(p/q)(r − c) + M (F) and F = F − M (F) 0 0 and recall (G.525) we ﬁnd the conjugation relation (G.514). −1/(10q) Note that sincewehaveassumedthat F ∈ O (e W ) satisﬁes (G.529)we σ h,D(c,ρ) −1/q have ∈ O (e W ) ∩ TC(2A, 2B) and the ﬁrst inequality of (G.515)issatisﬁed. σ h,D(c,ρ) The other inequalities of (G.515) are consequences of (G.526)and (G.527)and (G.516)is a consequence of (G.528), (G.524)and (G.543). Remark G.2. — Notice that from the ﬁrst inequality of (5.111) in Lemma 5.2 res (G.544) F F . W W −1/q −1/q h,D(c,ρ) e h,D(c,e ρ) Appendix H: Approximations by vector ﬁelds The main result of this Section is the following proposition on the approximation of an exact symplectic diffeomorphism close to an integrable one by a vector ﬁeld. Proposition H.1. — There exists a constant C > 0 for which the following holds. Let 0 <ρ < 1, F ∈ O (T × D(0,ρ)) and ∈ O (D(0,ρ)), (r) = O(r ).If ρ> 0 is small σ h σ 1/3 enough, h ρ and −9 (H.545) C × (ρh) F < 1 h,ρ 162 RAPHAËL KRIKORIAN then, there exist ∈ O (D(0,ρ/2)), A (F) ∈ O (T × D(0,ρ/2)) such that σ 3 σ h/2 (H.546) ◦ f = ◦ f F A (F) with 1/4 (H.547) = + F ◦ + O(ρ F ) −/2 h,ρ 1/4 (H.548) = + F + O(ρ F ) h,ρ and −1/4 (H.549) A (F) < exp(−ρ )F . 3 h/2,ρ/2 h,ρ The proof of this proposition is given in Section H.2. H.1 Auxiliary result. Proposition H.2. — Let ρ> 0, (r) = O(r ), ∈ O (D(0,ρ)), F, G ∈ O (T × σ σ h D(0,ρ)) such that (C some universal constant) −4 (H.550)C × (ρδ) (F +G )< 1. h,ρ h,ρ 1/3 −δ Then for any h/2 >δ ρ , there exists A(F, G) ∈ O (e (T × D(0, ρ))) such that σ h (H.551) = ◦ ◦ f +F+G +F G◦ A(F,G) /2 with −4 −3 −δ (H.552) A(F, G) (ρδ) (F +G ) + ρδ G . h−δ/2,e ρ h,ρ h,ρ h,ρ Proof. — To simplify the notations we denote W = W and we assume that h,D(0,ρ) ω(r) := ∇(r), ω(0) = 0satisﬁes ω(r) = r + O(r ). If −2 (δρ) max(F ,G )< 1 h,ρ h,ρ −2δ t t t the images of the domain e Wby the ﬂows , , ,0 ≤ t ≤ 1, are contained +F +F+G −δ in e W. Let us denote σ := max(DF ,DG ) h,δ h,δ ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 163 −2δ and for x = (θ , r) ∈ e Wand t ∈[−1, 1] t t (t, x) = (x) − (x). +F+G +F By classical theorems on ODE’s for t ∈[−1, 1] t t (t,·) = O(σ ), − = O(σ ). +F On the other hand one has t t (H.553) (t, x) = J∇( + F + G) ◦ (x) − J∇( + F) ◦ (x) +F+G +F dt = (I)(t, x) + (II)(t, x) + (III)(t, x) with t t (I)(t, x) = J∇ ◦ (x) − J∇ ◦ (x) +F+G +F t t (II)(t, x) = J∇ F ◦ (x) − J∇ F ◦ (x) +F+G +F (III)(t, x) = J∇ G ◦ (x). +F+G t t Since (x) = (x)+ ( (t, x), (t, x)) and − = O(σ ) one has (note θ r +F +F+G +F that r ◦ = r + O(σ )) +F t t ω(r ◦ (x) + (t, x)) − ω(r ◦ (x)) +F +F (H.554) (I)(t, x) = ∂ω(r) (t, x) + O(σ| (t, x)|) r r and (H.555) |(II)(t, x)|= O(|D F||(t, x)|). We have using the fact that − = O(σ ) and ω(r) = O(r) +F+G (H.556) (III)(t, x) = J∇ G(θ + tω(r), r) + O(εD G) 2 2 ∂ G(θ , r) + tω(r)∂ G(θ , r) + O(ρ ∂ ∂ G) r r θ r θ 2 3 −∂ G(θ , r) − tω(r)∂ G(θ , r) + O(ρ ∂ G) θ θ + O(σD G). 164 RAPHAËL KRIKORIAN Summing (H.554), (H.555), (H.556) and integrating (H.553)gives (t, x) ∂ω(r) (s, x)ds θ r (H.557) = (t, x) 2 2 t∂ G(θ , r) + (t /2)ω (r)∂ G(θ , r) θ r 2 2 −t∂ G(θ , r) − (t /2)ω (r)∂ G(θ , r) + O(ε +|D F|) |(s, x)|ds + A with 2 2 2 A = O(σD G) + O(ρ D∂ G). Lemma H.3. — One has 2 2 2 |(t, x)|≤ A := O(DG+ ρD∂ G)+ O(σD G)+ O(ρ D∂ G). 2 θ Proof. — From (H.557) and the fact that ∂ω(r) 1 |(t, x)|≤ C(1+ε+D F ) |(s, x)|ds+ O(DG+ ρD∂ G)+ A h,ρ θ 1 and we conclude by Grönwall inequality. Looking at the second component of (H.557)gives (ω(r) = O(r)) 2 2 (t, x)=−t∂ G(θ , r) + O(ρ∂ G) + O((ε +D F)A ) + A r θ 2 1 hence (integrating again and putting the result in (H.557)) (t, x) (H.558) (t, x) 2 2 −∂ω(r)(t /2)∂ G(θ , r) + t∂ G(θ , r) + (t /2)ω (r)∂ G(θ , r) θ r θ r −t∂ G(θ , r) − (t /2)ω (r)∂ G(θ , r) + O(A ) with 2 2 A = O(ρ∂ G) + O((ε +D F)A ) + A . 3 2 1 Taking t = 1gives (x) +F+G = (x) +F ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 165 −(∂ ω (r)/2)∂ G(θ , r) + ∂ G(θ , r) + (ω (r)/2)∂ G(θ , r) θ r θ r −∂ G(θ , r) − (ω (r)/2)∂ G(θ , r) + O(A ). On the other hand ∂ (G(θ − ω(r)/2, r)) −∂ (G(θ − ω(r)/2, r)) −(∂ ω (r)/2)∂ G(θ − ω(r)/2, r) + ∂ G(θ − ω(r)/2, r) θ r −∂ G(θ − ω(r)/2, r) hence J∇(G ◦ ) ◦ −/2 −(∂ ω (r)/2)∂ G(θ + ω(r)/2, r) + ∂ G(θ + ω(r)/2, r) θ r −∂ G(θ + ω(r)/2, r) and from Taylor Formula and the fact that ω(r) = O(r) (x) − (x) = J∇(G ◦ ) ◦ + O(ρ∂ G) + O(A ). +F+G +F −/2 3 Since = + O(σ ), this means that +F = (id + J∇(G ◦ ) ◦ + O(σD G) + O(A ) +F+G −/2 +F 3 thus = ◦ + O(A ) +F+G G◦ +F 3 −/2 or = f ◦ ◦ +F+G O(A ) G◦ +F 3 −/2 with −2 2 −1 −2 −2 A ((ρδ) σ + ρ (ρδ) δ + (σ + (ρδ) σ) −1 −1 −1 −2 × ((ρδ) + ρ(ρδ) δ ) + ρδ )G h,ρ −3 −3 ((ρδ) σ + ρδ )G h,ρ −4 −3 (ρδ) (F +G ) + ρδ G h,ρ h,ρ h,ρ provided −4 −3 (ρδ) (F +G )< 1,ρδ < 1. h,ρ h,δ 166 RAPHAËL KRIKORIAN To conclude we observe that if we apply the preceding formula with − and −F instead of ,F = f ◦ ◦ −−F−G O(A ) −G◦ −−F 3 /2 and inverting = ◦ + f . +F+G +F G◦ O(A ) /2 3 Corollary H.4. — Under the same conditions of Proposition H.4 one has ◦ f = ◦ f +F G +F+G◦ A (F,G) −/2 2 with −4 −3 −δ (H.559) A (F, G) (ρδ) (F +G ) + ρδ G . 2 h−δ/2,e ρ h,ρ h,ρ h,ρ Proof. — If we apply (H.551)with G ◦ instead of G we get −/2 = ◦ ◦ f +F+G◦ +F G A(F,G◦ ) −/2 −/2 hence −1 −1 ◦ f = ◦ f ◦ ◦ f +F G +F+G◦ G −/2 A(F,G◦ ) G −/2 = ◦ f +F+G◦ A (F,G) −/2 2 where A (F, G) = A(F, G ◦ ) + O(|DG||D G|) satisﬁes (H.559)(cf. (H.552)). 2 −/2 3/2 H.2 Proof of Proposition H.1.— Let δ = c/(k + 1) , h = h − δ /2, ρ = (3/4)ρ , k k k 0 −δ ρ = e ρ and c chosen such that h ≥ h/2, ρ ≥ ρ/2for all k ∈ N. Using Corollary H.4 k k k we construct sequences S , G such that S = 0, G = F k k 0 0 (H.560) ◦ f = ◦ f +S G +S G k k k+1 k+1 S = S + G ◦ k+1 k k −/2 (H.561) G = A (S , G ) k+1 2 k k with S ≤S +G k+1 h ,ρ k h ,ρ k h ,ρ k+1 k+1 k k k k and −3 −4 (H.562) G ρ δ G + (ρ δ ) (S +G )G k+1 h ,ρ k k h ,ρ k k k h ,ρ k h ,ρ k h ,ρ k+1 k+1 k k k k k k k k k ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 167 as long as −4 (ρ δ ) (S +G )< 1. k k k h ,ρ k h ,ρ k k k k With ε =G and σ := S we have (s = 0) n n h ,ρ n n h ,ρ 0 n n n n −3 −4 (H.563) ε ≤ C(ρ δ + (ρ δ ) ε )ε k+1 k k k j k j=0 (H.564) σ ≤ σ + O(ε ) k+1 k k −4 as long as (ρ δ ) (s + σ )< 1. k k k k Let k be the largest integer for which these sequences are deﬁned. We observe 1−3/4 −3 ∗ 1/4 −1 −2 1/4 that for k < min(k ,ρ ) one has (ρ δ ) ≤ ρ and ρ δ ≤ ρ = ρ ;hence,if k k k k k ∗ −1/4 k = min(k ,ρ ) one has 1/4 −9 ∀ k < k,ε ≤ Cρ (1 + ρ ε )ε . k+1 j k j=0 We are in position to apply Lemma F.1 with α = ρ , θ = 1/4, a = 9: since condition (F.483)issatisﬁed(cf. (H.545)) one has ∗ 1/4 ∗ k/4 k ≥ ρ , ∀ k ≤ k ,ε ≤ (2Cρ) ε k 0 and also 1/4 s − ε ≤ ε ρ ε . k 0 j 0 j=1 To conclude the proof we set = S , A (F) = F . k k Appendix I: Adapted KAM domains: lemmas I.1 Proof of Lemma 10.1. I.1.1 Proof of the RHS inequality of (10.306). — From (7.193) and the deﬁnition (10.300)of i (ρ),for every (k, l) ∈ E ,0 < k < N ,0≤|l|≤ N one has − i (ρ)−1 i (ρ)−1 i (ρ)−1 − − − (i (ρ)−1) − −1 D(c , K ) ∩ D(0, 2ρ)=∅, l/k i (ρ)−1 (i (ρ)−1) (i (ρ)−1) − − hence |c | >ρ . Since ω (c ) = l/k,wededucefromthe fact that l/k i (ρ)−1 l/k (i (ρ)−1) satisﬁes a (2A, 2B)-twist condition (7.166)that |(l/k) − ω |=|ω(c ) − ω(0)|≥ l/k 168 RAPHAËL KRIKORIAN −1 (2A) ρ . By Dirichlet Approximation Theorem, for any L ≥ 1 there exist k, l ∈ Z,0 < |k|≤ Lsuch that |ω − (l/k)|≤ ,hence |k|L −1 (I.565) (2A) ρ ≤ . |k|L 1 1 However, since ω ∈ DC(τ ) one has |ω − (l/k)|≤ hence 0 1+τ 0 |k| |k|L 1/τ L |k|. This and (I.565)yield −(1+1/τ ) ρ L . In particular if one chooses L = N − 1 N one gets i (ρ)−1 i (ρ) − − −(1+1/τ ) ρ N i (ρ) which proves the inequality of the RHS of (10.306). I.1.2 Proof of the LHS inequality of (10.306). — Let us prove the second inequality of (10.306). By deﬁnition of i (ρ) there exists (l, k) ∈ Z ,0 < k < N , |l|≤ N such − i (ρ) i (ρ) − − that (cf. (7.193)) (i (ρ)) − −1 D(c , 2K ) ∩ D(0, 2ρ) = ∅. l/k i (ρ) (i (ρ)) − 1+τ In particular (cf. (7.163)) |c |≤ 3ρ . Since ω ∈ DC(κ, τ ), |ω(0)− (k/l)|≥ κ/k and l/k (i (ρ)) − 1/2 1+τ from (7.192) |ω (0) − ω (c )|≥ κ/k − 2ε ; by the twist condition 6Aρ ≥ i (ρ) i (ρ) l/k − − (i (ρ)) − −(1+τ) 2A|c |≥ κ N − ρ hence l/k i (ρ) −(1+τ) ρ N i (ρ) which shows that the LHS of (10.306) holds. Estimates (10.304), (10.305) are then immediate. I.2 Proof of Items 1, 2, 4 of Proposition 10.2.— Recall that from (10.307) ((3/2)ρ) i−1 (j) −1 U = D(0,(3/2)ρ) D(c , s K ), j,i−1 i j=1 (k,l)∈E l/k j (I.566) i−1 m=j s = e ∈[1, 2] j,i−1 (j) where E ⊂{(k, l) ∈ Z , 0 < k < N , 0 < |l|≤ N }, ω (c ) = l/k. In particular, any j j j j l/k (j) −1 D ∈ D(U ) is of the form D = D(c , s K ),where j ≤ i − 1, (k, l) ∈ E . i j,i−1 j l/k j Lemma I.1. — If D ∈ D (U ) then j ≥ i (ρ). (3/2)ρ i − ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 169 Proof. — Since D(0, 2ρ) = D(0, 2ρ) ∩ U ,from(I.566)for all j ≤ i (ρ) − 1, i (ρ) − (j) −1 (k, l) ∈ E one has |c |≥ 2ρ + K . On the other hand, if D ∈ D(U ) is of the form j i l/k j (j) −1 D = D(c , s K ),where j ≤ i − 1, (k, l) ∈ E and intersects D(0,(3/2)ρ) one has j,i−1 j l/k j (j) −1 −1 |c |≤ (3/2)ρ + 2K < 2ρ + K hence j ≥ i (ρ). l/k j j From (7.193) and Lemma I.1 we can thus write (j) ((3/2)ρ) i−1 −1 U = D(0,(3/2)ρ) D(c , s K ), j,i−1 i l/k j j=i (ρ) (k,l)∈E − j (I.567) i−1 m=j s = e ∈[1, 2]. j,i−1 We deﬁne i−1 Q = {l/k,(k, l) ∈ E } i j j=i (ρ) and for t ∈ Q j(t, i) = min{j : j ∈ N∩[i (ρ), i − 1],(k, l) ∈ E and l/k = t} − j (j(t,i)) c(t, i) = c , s(t, i) = s . j(t,i),i−1 Deﬁne for i (ρ) ≤ j < i ≤ i (ρ), − + −1 κ = s K . j,i j,i−1 We observe that from the inequality N ≤ N ,for any i (ρ) ≤ j < i ≤ i (ρ), i (ρ) − + + i (ρ) i (ρ) ≤ j < i ≤ i (ρ), (i, j) = (i , j ), one has − + −2 1/2 κ + κ N , ε |κ − κ |; j,i j ,i j,i j ,i max(j,j ) min(j,j ) if j < j or j = j and i < i , one has κ <κ hence from Lemma 7.3,Item(2)for j,i j ,i (k, l) ∈ E , (k , l ) ∈ E one has j j (j) (j ) either l/k = l /k and D(c ,κ ) ∩ D(c ,κ )=∅ j,i j ,i l/k l /k (I.568) (j) (j ) or l/k = l /k and D(c ,κ ) ⊂ D(c ,κ ). j,i j ,i l/k l /k As a consequence, for j ∈ N ∩[i (ρ), i − 1], (k, l) ∈ E , t = l/k, one has the inclusion − j (j) −1 −1 D(c , s K ) ⊂ D(c(t, i), s K ), and therefore (cf. (I.567), (10.308)) j,i−1 j(t,i),i−1 l/k j j(t,i) ((3/2)ρ) −1 U = D(0,(3/2)ρ) D(c(t, i), s K ). j(t,i),i−1 i j(t,i) t∈Q 1/2 46 −1 This is clear if j = j ;if j = j observe that if i = i , ε |s − s |K . j,i−1 j,i −1 i (ρ) j − 170 RAPHAËL KRIKORIAN This implies that any D ∈ D (U ) is of the form (3/2)ρ i −1 (I.569)D = D(c(t, i), s K ), t ∈ Q , j(t, i) ≤ i − 1. j(t,i),i−1 i j(t,i) Proof of item 1 of Proposition 10.2.— This is a consequence of (I.569)and (I.568). Proof of item 2 of Proposition 10.2.— One can write for some t ∈ Q , t ∈ Q ,D = i i −1 −1 D(c(t, i), s K ),D = D(c(t , i ), s K ) and from Lemma 7.3,Item(2)if j(t,i),i−1 j(t ,i ),i −1 j(t,i) j(t ,i ) D ∩ D = ∅ one has t = t . On the other hand since t = t ∈ Q ⊂ Q one has j(t, i ) = i i j(t, i).Wenow usethe fact that s ≤ s . j(t,i ),i −1 j(t,i),i−1 Proof of item 4 of Proposition 10.2.— Let us prove that D ∈ D (U ) is a subset of ρ i (ρ) U . If this were not the case, there would exist D ∈ D(U ) such that D ∩ D = ∅;in i i D D particular D ∈ D (U ) andfromitem 2 D ⊂ D; but this contradicts the deﬁnition (3/2)ρ i of i .Hence D ⊂ U . D i This latter inclusion and (I.567) applied with i = i show that one has D ∩ (j) −1 D(c , s K ) =∅ for all i (ρ) ≤ j ≤ i − 1, (k, l) ∈ E . As a consequence D = j,i−1 − D j l/k j (j) −1 D(c , s K ) for some j ≥ i , (k, l) ∈ E . j,i (ρ)−1 D j l/k + j On the other hand, by deﬁnition of i there exists D ∈ D (U ) of the form D ρ i +1 −1 D = D(c , s K ) with j ≤ i , s ∈[1, 2] (cf. (I.569)) such that D ⊂ D. One hence have −1 −1 s K ≥ K thus j ≥ i . We conclude that j = i . j,i (ρ)−1 D D + j i Appendix J: Classical KAM measure estimates J.1 A lemma. — If A and B are two sets we denote by A"B = (A ∪ B) (A ∩ B) their symmetric difference. Lemma J.1. — Let A = I I ,where I ⊂ R is an interval and all the intervals I are j j j∈J 1/2 1 disjoint. Then if |I | ≤ 1 and if g : M → M is a C -symplectic diffeomorphism such that j R R j∈J 1/2 g − id 1 ≤ 1/10, then one has Leb(W " W ) g − id . C A g(A) 0 Proof. — We can assume that the intervals I are contained in I. Recall that 1 =|1 − 1 | and notice that since the intervals I are pairwise disjoint one has A " B A B j 1 = 1 − 1 hence W W W A I I j∈J 1 = χ − χ W " g(W ) j A A j∈J ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 171 where χ = 1 − 1 , χ = 1 − 1 . This gives W g(W ) j W g(W ) I I I I j j Leb (W " g(W )) M A A =χ − χ 1 j L j∈J ≤χ 1 + χ 1 L j L j∈J ≤ Leb (W " g(W )) + Leb (W " g(W )). M I I M I I R R j j j∈J On the other hand, if I is an interval, there exist intervals I ⊂ I ⊂ Isuch thatW ⊂ −1 g(W ) ⊂ W and max(|I " I|,|I " I||) ≤ 2max(g − id 0 ,g − id 0 ) ≤ Cg − id 0 , I I C C C C > 0 depending only on M (recall that we have assumed g − id 1 is small enough). This is clear in the (AA)-case and in the (CC) or (CC*)-case it follows from the (AA)-case using the symplectic changes of coordinates ψ and ϕ (4.79), (4.77). Therefore, since g is symplectic, Leb (W " g(W )) ≤ Cmin(g − id 0 , Leb (W )). M I I C M I R j j R j 1/2 1/2 In particular Leb (W " g(W )) ≤ Cg− id Leb (W ) and since Leb (W ) ≤ M I I 0 M I M I R j j R j R j |I | the conclusion follows. J.2 Proof of Theorem 12.1.— We use the notations of Section 7 and Propositions 7.1, 7.2, 7.5 and Remark 7.1. We apply Proposition 7.5–Remark 7.1 with m = 1 and Proposition 4.3 with A = −2δ e U, L = L , A = U = U, 1,Prop. 7.5 1 Leb (W −2δ L(f , W )) M 1 R R∩e U R∩U 1/2 −2δ ≤ C × (Leb (R ∩ (e U L))+g − id ) R 1,∞ 0 1 1 1/8 2(a +3) 2(a +3) 0 0 ε + ε ε . Appendix K: From (CC) to (AA) coordinates Sometimes we need to reduce the (CC)-case to the (AA)-case, for example when deﬁning the Hamilton-Jacobi Normal Form in Section 8 or in Section 16. For α ∈]0,π[ deﬁne the angular sector (ρ) ={r ∈ D(0,ρ), arg(r)/∈[−α, α]} − + and (ρ)=− (ρ). Recall the deﬁnition of the maps ψ , cf. (4.79)ofSection 4.1. α α 172 RAPHAËL KRIKORIAN CC CC h/2 2 CC Lemma K.1. — Let c ∈ R, F ∈ O (W ), ε = Ce D F CC , h,D(c,2ρ) W h,D(c,2ρ) −2 −2 (K.570)Cδ ρ ε< 1 and α ∈]δ, π − δ[. CC 3 AA (1) If c = 0 and F = O (z,w) there exists F ∈ O (T × (ρ − 4δ)) such that on σ h−δ ± α+4δ T × (ρ − 4δ) one has h−4δ α+4δ −1 AA CC CC AA CC (K.571) f = ψ ◦ f ◦ ψ , F = F ◦ ψ + O (F ). ± ± 2 F F ± ± AA (2) If |c| > 4ρ , there exists F ∈ O (T × D(0,ρ)) such that (K.571) holds. σ h−δ Proof. — We prove item (1), the proof of item (2)isdoneinasimilar(andeven h−δ simpler) way. From ε ≤ δ and f (0) = 0we get that if z,w satisfy |z|,|w| < e (ρ − 1/2 ± 3δ) , r =−izw ∈ (ρ− 3δ) then z, w deﬁned as ( z, w) = f CC (z,w) satisfy | z− z|≤ α+3δ −h/2 −h/2 ε h 1/2 ε h 1/2 e ε|z|, |w − w|≤ e ε|w| and thus | z|≤ e |z|≤ e ρ and |w |≤ e |w|≤ e ρ ;on the other hand if r =−i zw one has z/w (K.572) | r − r|≤ 3ε|r|, | arg( r) − arg(r)|≤ 3ε, − 1 ≤ (5/2)ε. z/w Since ε< δ, ± ± CC f ◦ ψ (T × (0,ρ − 3δ)) ⊂ ψ (T × (0,ρ)) ± h−3δ ± h α+3δ α AA −1 ± hence f := ψ ◦ f CC ◦ ψ : T × (0,ρ − 3δ) → T × (0,ρ) is well deﬁned. F ± h−3δ h ± α+3δ α −1 AA On the other hand if ψ (z,w) = (θ , r), f (θ , r) = (θ, r), ψ (θ, r) = ( z, w) one has ± ± from (K.572) and Lemma M.1 max(|θ − θ| ,| r − r|) ≤ 3ε 2π Z hence AA f − id ≤ 3ε T × (0,ρ−3δ) h−3δ α+3δ AA and from Remark 4.2, Lemmata 4.4, 4.5 and condition (K.570) there exists F ∈ ± AA AA O(T × (0,ρ − 4δ)) such that f = f and h−4δ α+4δ AA AA (K.573) f = φ ◦ f . AA F O (F ) ± J∇ F ± To get the second estimate in (K.571)wenoticethat CC CC f = φ ◦ f CC F O (F ) J∇ F 2 ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 173 hence −1 1 f AA = ψ ◦ φ ◦ f CC ◦ ψ CC F O (F ) ± ± J∇ F 2 1 −1 = φ ◦ ψ ◦ f CC ◦ ψ CC O (F ) ± J∇(F ◦ψ ) ± 2 = φ ◦ f CC CC O (F ) J∇(F ◦ψ ) 2 and from (K.573) AA CC CC F = F ◦ ψ + O (F ). ± 2 CC CC Remark K.1. —If f = ◦ f CC we have (cf. Section 4.2). −1 CC AA AA ψ ◦ f ◦ ψ = ◦ f . Appendix L: Lemmas for Hamilton-Jacobi Normal Forms L.1 Proof of Lemma 8.3.— Since ∂ (r) 1(cf. (8.218)), (8.222)and (8.220)show that there exists e : T → C, e ∈ O (T ) such that 0 qh/3 0 σ qh/3 −1 2 (L.574) ∀ θ ∈ T ,∂ (θ , e (θ )) = 0, e (qρ) q ε. qh/3 r 0 0 T qh/3 We now make a Taylor expansion: using (L.574) we see that (L.575) (θ , r) = (θ , e (θ ) + (r − e (θ )) 0 0 2 2 = (θ , e (θ )) + (1/2)∂ (θ , e (θ ))(r − e (θ )) 0 0 0 3 k k−3 + (r − e (θ )) ∂ (θ , e (θ ))(r − e (θ )) 0 0 0 k! k=3 and if we deﬁne (L.576) (θ) = (1/2)∂ (θ , e (θ )), e (θ )=−(θ , e (θ ))/ (θ ) 0 1 0 one gets for some p(θ , r) 2 3 (θ , r) = (θ) −e (θ ) + (r − e (θ )) + (r − e (θ )) p(θ , r) 1 0 0 = (θ , r − e (θ )) −1 −2/q with ∈ O(T × D(0, e qρ/2 − Cqρ ε)) ⊂ O(T × D(0,ρ )) qh/3 qh/3 q 2 3 (θ , r) = (θ) r − e (θ ) + r p(θ , r + e (θ )) ; 1 0 this gives the desired form for (θ , r) if one sets f (θ , r) = p(θ , r + e (θ )). 0 174 RAPHAËL KRIKORIAN The estimates (8.228)on e , e , are then clear from (L.574), (L.576). Let us check 0 1 the one on f .From(8.223)and (8.227)wehave 2 3 (θ)(r − e (θ ) + r f (θ , r)) 2 i 3 =: r + f (θ )(r + e (θ )) + r (b(r) + f (θ , r + e (θ ))) i 0 0 i=0 hence from (8.225) and the ﬁrst two inequalities of (8.228) 3 −1 −3 2 r f (θ , r) − (θ) b(r) − f (θ , r + e (θ )) (qρ) q ε and by the maximum principle −1 sup f (θ , r) − (θ) b(r) − f (θ , r + e (θ ))) (θ ,r)∈T ×D(0,ρ ) qh/3 q −3 −3 2 ρ (qρ) q ε 1. We the conclude by (8.218)and (8.225). L.2 Square roots. ∗ 2 1/2 Lemma L.1. — Let a ∈ C . There exists a unique function m (z) = z(1+ a/z ) univalent 1/2 on C D(0,|a| ) such that 2 2 −1 (L.577) m (z) = z + a, m (z) = z + O(z ). 1/2 It satisﬁes for z, z ∈ E := {w ∈ C,|w| > L|a| } (L > 3) 2 m (z) − m (z ) 2 a a −2/L 1/L (L.578) (2/π )e ≤ ≤ (π/2)e z − z 2 1/2 Proof. — The existence and uniqueness of m (z) = z(1 + (a/z )) is clear. Note that the inverse for the composition of m is m and that if L > 2 m (E ) ⊂ a −a a L 2 −1/2 E . On the other hand the derivative of m (z) is equal to ∂ m (z) = (1 + a/z ) and 3L/4 a z a −1/2 1/L since for t ∈[0, 1/2], (1 − t) ≤ 1 + t onegetsfor z ∈ E (L > 2) |∂ m (z)|≤ e . L z a Now any two points z, z ∈ E can be joined by a path in E the length of which is L L 1/L ≤ (π/2)|z − z |;thusfor any z, z ∈ E , |m (z) − m (z )|≤ (π/2)e |z − z | which is L a a the right hand side inequality of (L.578). To get the left hand side we use the fact that 1/(3L/4) |m (m (z)) − m (m (z ))|≤ (π/2)e |m (z) − m (z )| if L > 3(3L/4 > 2). −a a −a a a a ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 175 2 1/2 L.3 Proof of Lemma 8.5.— From Lemma L.1 z → (z + a) is well deﬁned on 1/2 C {|z| > |a| }. −1/2 Let 0 ≤ s ≤ h/3. We are looking for g(θ , z) = (θ) z(1 + g ˚(θ , z)) such that 2 2 −1 2 z = (θ) z (θ) (1 + g ˚(θ , z)) − e (θ ) 3 −3/2 3 + z (θ) (1 + g ˚(θ , z)) f (θ , g(θ , z)) which can be written as a Fixed Point problem 1/2 e (θ ) −1/2 3 (L.579) g ˚(θ , z)= 1 + (θ) − z(θ) (1 + g ˚(θ , z)) f (θ , g(θ , z)) − 1. Using the estimate on f given by (8.228) one can see that the map " : g ˚ → R.H.S. of −2 (L.579)deﬁnesa 2ρ -contracting map on the ball B(0, CL ) of center 0 and radius −2 −1 CL of the Banach space (O(T × A(λ ,ρ )),· ) provided L and ρ are small sq s,L q ∞ q enough. By the Contraction Mapping Theorem it has a unique ﬁxed point g ˚ in this ball. In other words (L.580) (θ , g(θ , z)) = z . The fact that g ∈ O(T × A(λ ,ρ )) is uniquely deﬁned shows that the various g found qs s,L q for different values of s must agree. Hence g is deﬁned on (T × A(λ ,ρ )). qs s,L q 0≤s≤1 −1 L.4 Proof of Lemma 8.6.— We look for H under the form H(z) = γ z(1 + H(z)). Equation (8.238) can be written as a Fixed Point problem: −1 ˚ ˚ (γ z(1 + H(z)))) (L.581) H(z)=− . −1 ˚ ˚ (1 + (γ z(1 + H(z)))) By Cauchy’s estimates for z ∈ A(λ ,ρ ) s,L q −2 |∂(z)|≤ L . dist(z,∂ A(λ ,ρ )) s,L q −1 −2 Hence if z ∈ A(2λ ,(1/2)ρ ) the map u → (γ zu) is 4L -Lipschitz on {(3/4) ≤ s,L q −2 |u|≤ 4/3} and the map " deﬁned by the R.H.S. of (L.581)is 4L contracting on the −2 ball {H ≤ 2L }. It admits thus a unique ﬁxed point in this ball. A(2λ ,(1/2)ρ ) s,L q Appendix M: Some other lemmas Lemma M.1. — Let z ∈ C 176 RAPHAËL KRIKORIAN 1) One has iz |e − 1|≥ min(1, min|z − 2π l|). l∈Z 2) If z ∈ R, iz |e − 1|≥ min|z − 2π l|). l∈Z iz Proof. — 1) Let η := e − 1. We can assume |η| < 1/2. We can thus deﬁne iz := k−1 k iz iz ln(1 + η) = (−1) η /k such that e = 1 + η = e . There thus exists l ∈ Z such k∈N that z = z − 2π l.But |z |=| ln(1 + η)|≤ 2|η|. 0 0 2) Just use the fact that for |w|≤ π , |2sin(w/2)|≥ (2/π )|w|. Lemma M.2. — Let f ∈ C (T) be such that for some δ ∈]0, 1[, μ> 0 2 0 (M.582) f ≤ δf + μ. L (T) C (T) Then, for some C > 0, −1 −h/(12δ ) (M.583) f 0 ≤ δ μ + e f . C (T) h Proof. — If iikθ (M.584) f (θ ) = f (k)e k∈Z is the Fourier expansion of f , one has for some C > 0and anyN ∈ N −hN (M.585) f ≤ |f (k)|+ e f C (T) h |k|≤N 1/2 −hN (M.586) ≤ (2N + 1) f + e f L (T) 1/2 −hN (M.587) ≤ (3N) (δf 0 + μ) + e f . C (T) h −2 1/2 If we choose N = δ /12 we have (3N) δ ≤ 1/2and −1 −h/(12δ ) (M.588) f ≤ δ μ + e f . C (T) h h ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS 177 Appendix N: Stable and unstable Manifolds N.1 The Stable Manifold Theorem. — Let (E,·) be a Banach space, M : E → Ean invertible linear continuous map. Let κ, δ > 0. We say that M is (κ, δ)-hyperbolic if there exist κ> 0 and continuous projectors P ,P satisfying id = P + P ,P P = P P = 0, s u E s u s u u s P MP = P MP = 0such that s u u s −1 −κ max(P MP ,(P MP ) ) ≤ e s s u u −1 max(P ,P ) ≤ δ . s u The spaces E := P E, ∗= s, u, are then M-invariant and are the stable and unstable ∗ ∗ spaces of the linear map M. We shall use the notations M = P MP , ∗= s, u. ∗ ∗ ∗ Let B(0,ρ) ⊂ E be the ball of center 0 and radius ρ> 0. Theorem N.1 (Stable/Unstable Manifold Theorem).— Assume that M is (κ, δ)-hyperbolic as above and let F : B(0,ρ) → E be C . Assume that −1 −1 (N.589) F(0)≤ C δκρ , DF 1 ≤ C δκ. C (B(0,ρ)) Then, if C is large enough (but universal) (1) The map x → Mx+ F(x) has a unique hyperbolic ﬁxed point x such that max(P x,P x) ≤ s u (ρ/4) (in particular, it is located in B(0,ρ/2)). (2) The local stable (resp. unstable) manifold s n W (x; M + F) := {y ∈ B(x,ρ/4), ∀ n ≥ 0,(M + F) (y) ∈ B(x,ρ/2)} loc u n (resp. W (x; M+ F) := {y ∈ B(x,ρ/4), ∀ n ≤ 0,(M+ F) (y) ∈ B(x,ρ/2)}) of the point loc xfor M + F is of the form {x + γ (x ), x ∈ E ∩ B(0,ρ/2)} (resp. {x + γ (x ), x ∈ s s,F s s s u u,F u u E ∩ B(0,ρ/2)})where γ : E → E (resp. γ : E → E )isamapofclass C and u s,F s u u,F u s −1 −1 Dγ ≤ CDF (δκ ) (resp. Dγ ≤ CDF (δκ ) ). s,F B(0,ρ) u,F B(0,ρ) −1 (3) If G satisﬁes also (N.589)thenfor ∗= s, u, Dγ − Dγ ≤ CD(F− G) (δκ ) . ∗,F ∗,G B(0,ρ) (4) If F(0) = 0, DF(0) = 0 then x = 0 and T W (0) = E , ∗= s, u. 0 ∗ loc Notice that the Theorem gives the same size for the domains of deﬁnition of γ , s,F γ . u,F N.2 Proof of Lemma 15.4.— Using the deﬁnition of f (θ , r) = (ϕ, R) one can H + ω see that f (θ , r) = (θ , r) if and only if H + ω (N.590) ∇ H (θ , r)+∇ ω(r) = 0 or equivalently 0 = a(0)θ + ∂ b(0)r 0 = ∂ b(0)θ + ( + ∂ a(0))r + ∂ a(0) + ∂ ω(r). r r r r 178 RAPHAËL KRIKORIAN Solving the ﬁrst equation and inserting it into the second yields ∂ b(0) (N.591) θ =− r a(0) ∂ a(0) + ∂ ω(r) r r (N.592) r =− . (∂ b(0)) 2 r + ∂ a(0) − a(0) We observe that, cf. (15.422), 2 2 2 −1 −2 −qh max(|∂ a(0)|,|(∂ b (0)) / a(0)|) q ν ρ e ε </10 r p/q r q p/q 2 −1 −qh 2 |∂ a(0)| q ρ e ε ,∂ ω(r) = O(r ) r p/q r p/q and deduce by a simple ﬁxed point theorem (in dimension 1) that (N.592) has a unique real solution r ∂ a(0); returning to (N.591) and using (cf. (15.423), (15.422)) 0 r 2 −qh 2 −1 −qh (N.593) a(0) = q ν e ε , |∂ b(0)| q ρ e ε q p/q r p/q p/q we conclude that (N.590) has also a unique solution (θ , r ) ∈ D(0,ρ ) 0 0 p/q −1 2 −2 −qh 2 −1 −qh |θ | ν q ρ ε e , |r | q ρ ε e 0 p/q 0 p/q q p/q p/q −8 −8 72 2 −qh −100 and in particular since ρ = max((c /4) , q ) (cf. (15.405)), q e = O(q ), ε ≤ p/q p/q p/q c (cf. (15.406)), ν qρ (cf. (15.409)), a ≥ 10, one has q p/q 3 p/q 5 5 2 (N.594) (θ , r ) ∈ (D(0,ρ ) × D(0,ρ )) ∩ R . 0 0 p/q p/q We now compute Df (θ , r ). Since ω depends only on the r-variable one has, cf. H + ω 0 0 (4.95) of Lemma 4.6, f = ◦ f H + ω ω H Q Q hence 1 ∂ ω(r ) Df (θ , r ) = Df . 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Soc., 36 (1934), 63–89. 56. N. T. ZUNG, Convergence versus integrability in Birkhoff normal form, Ann. Math., 161 (2005), 141–156. R. K. Department of Mathematics, CNRS UMR 8088, CY Cergy Paris Université (University of Cergy-Pontoise), 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise, France raphael.krikorian@cyu.fr Manuscrit reçu le 29 juin 2020 Version révisée le 22 décembre 2021 Manuscrit accepté le 25 janvier 2022 publié en ligne le 27 avril 2022.

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On the divergence of Birkhoff Normal Forms

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On the divergence of Birkhoff Normal Forms

On the divergence of Birkhoff Normal Forms

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