Hyperbolicity of renormalization of critical circle maps

Michael Yampolsky

doi:10.1007/s10240-003-0007-1

Hyperbolicity of renormalization of critical circle maps

Yampolsky, Michael 2003-05-01 00:00:00 HYPERBOLICITY OF RENORMALIZATION OF CRITICAL CIRCLE MAPS by MICHAEL YAMPOLSKY The renormalization theory of critical circle maps was developed in the late 1970’s–early 1980’s to explain the occurence of certain universality phenomena. These phenomena can be observed empirically in smooth families of circle home- omorphisms with one critical point, the so-called critical circle maps, and are analo- gous to Feigenbaum universality in the dynamics of unimodal maps. In the works of Ostlund et al. [ORSS] and Feigenbaum et al. [FKS] they were translated into hyperbolicity of a renormalization transformation. The ﬁrst renormalization transformation in one-dimensional dynamics was constructed by Feigenbaum and independently by Coullet and Tresser in the set- ting of unimodal maps. The recent spectacular progress in the unimodal renormal- ization theory began with the seminal work of Sullivan [Sul1,Sul2,MvS]. He intro- duced methods of holomorphic dynamics and Teichmul¨ ler theory into the subject, developed a quadratic-like renormalization theory, and demonstrated that renor- malizations of unimodal maps of bounded combinatorial type converge to a horse- shoe attractor. Subsequently, McMullen [McM2] used a different method to prove a stronger version of this result, establishing, in particular, that renormalizations converge to the attractor at a geometric rate. And ﬁnally, Lyubich [Lyu4,Lyu5] constructed the horseshoe for unbounded combinatorial types, and showed that it is uniformly hyperbolic, with one-dimensional unstable direction, thereby bringing the unimodal theory to a completion. The renormalization theory of circle maps has developed alongside the uni- modal theory. The work of Sullivan was adapted to the subject by de Faria, who constructed in [dF1,dF2] the renormalization horseshoe for critical circle maps of bounded type. Later de Faria and de Melo [dFdM2] used McMullen’s work to show that the convergence to the horseshoe is geometric. The author in [Ya1,Ya2] demonstrated the existence of the horseshoe for unbounded types, and studied the limiting situation arising when the combinatorial type of the renormalization grows without a bound. Despite the similarity in the development of the two renormalization theories up to this point, the question of hyperbolicity of the horseshoe attractor presents a notable difference. Let us recall without going into details the structure of the argument given by Lyubich in the unimodal case. The ﬁrst part of Lyubich’s work was to to endow the This work was partially supported by NSF Grant DMS-9804606. 2MICHAELYAMPOLSKY ambient space of the renormalization transformation with the structure of a complex- analytic manifold, with respect to which renormalization is an analytic operator. He then showed that the stable sets of periodic points of this operator are codimension one submanifolds and used an argument based on McMullen’s Tower Rigidity Theorem and an inﬁnite dimensional version of the Schwarz Lemma to show that renormaliza- tion is a strict contraction in the stable direction. The second part of Lyubich’s work was to show the existence of an unstable direction. This is a fundamentally more dif- ﬁcult part of the proof, based on Lyubich’s Rigidity Theorem [Lyu3] and an elegant argument relating non-hyperbolicity of the renormalization operator to certain topo- logical properties of its orbits. In contrast, in the case of critical circle maps it has been well-known how one may construct an unstable direction for the renormalization. The difﬁculty arises with the ﬁrst part of the program. There is no obvious way to turn the renormalization into an analytic operator. Indeed, the deﬁnition of renormalization for critical circle maps essentially employs real symmetry, and resists complexiﬁ- cation. To circumvent this obstacle, in this paper we introduce a different renor- malization construction, a renormalization of critical cylinder maps. This construction is strongly motivated by a so-called parabolic renormalization, a degenerate case of renormalization, which we studied in detail in [Ya2]. There is a natural functorial relation between the renormalization of cylinder maps and the usual one which allows us to transfer the results back and forth. By virtue of its construction, the cylinder renormalization is an analytic operator on a Banach manifold. We then show that the stable sets of the periodic orbits of this operator are codimension one submanifolds, and prove that every such orbit is hyperbolic. Our result gives a rigorous mathematical explanation of the universality phenomena, settling this well-studied problem. The structure of the paper is as follows. In §1weintroducecritical circle maps. We discuss the universality phenomena and deﬁne renormalization in §2. In §4 we formulate the Renormalization Hyperbolicity Conjecture (Lanford’s Program) and state our results. In the following section we discuss some of the constructions from de Faria’s and our own earlier work. In §6 we introduce parabolic renormal- ization, mainly as a motivation for futher discussion. We then deﬁne the cylinder renormalization operator in §7 and proceed to prove our main theorems. Acknowledgements I had many fruitful discussions with Adam Epstein on the subject of parabolic renormalization of critical circle maps and related topics, and I wish to thank him for generously sharing his mathematical insights with me. I am also grateful to Folkert Tangerman for explaining to me the construction of the unstable direc- HYPERBOLICITY OF RENORMALIZATION 3 tion some years ago. I would like to express my gratitude to Mikhail Lyubich for the many conversations we had as this paper developed, for his advice, patience, and constant encouragement. I want to thank Welington de Melo for the helpful criticism of an earlier version of this work. The bulk of this work was conducted at the Yale Mathematics Department, and I wish to thank my colleagues there for making my postdoctoral experience so enjoyable. My special thanks go to Pe- ter Jones for his guidance. Thanks are also due to IHES where this paper was completed. Finally, I wish to thank the referee for the helpful comments. 1. Preliminaries Some notations. — We use dist and diam to denote the Euclidean distance and diameter in C. We shall say that two real numbers A and B are K-commensurable −1 for K>1 if K |A|≤ |B|≤ K|A|. The notation D (z) will stand for the Euclidean disk with the center at z ∈ C and radius r. The unit disk D (0) will be denoted D. The plane (C \ R) ∪ J with the parts of the real axis not contained in the interval J ⊂ R removed will be denoted C .By the circle T we understand the afﬁne manifold R/Z, it is naturally identiﬁed with the unit circle S = ∂D.The real translation x → x + θ projects to the rigid rotation by angle θ , R : T → T.For two points a and b in the circle T which are not diametrically opposite, [a, b] will denote the shorter of the two arcs connecting them. As usual, |[a, b]| will denote the length of the arc. For two points a, b ∈ R, [a, b] will denote the closed interval with endpoints a, b without specifying their order. The cylinder in this paper, unless otherwise speciﬁed will mean the afﬁne manifold C/Z. Its equator is the circle {Im z = 0}/Z ⊂ C/Z. A topological annulus A ⊂ C/Z will be called an equatorial annulus, or an equatorial neighborhood, if it has a smooth boundary and contains the equator. By “smooth” in this paper we will mean “of class C ”, unless another de- gree of smoothness is speciﬁed. The notation “C ” will stand for “real-analytic”. Critical circle maps. — A critical circle map is an orientation preserving auto- morphism of T of class C with a single critical point c. A further assumption is made that the critical point is of cubic type. This means that for a lift f : R → R of a critical circle map f with critical points at the integer translates of ¯ c, ¯ ¯ f (x) − f (¯ c) = (x − ¯ c) (const + O(x − ¯ c)). We note that all the renormalization results will hold true if in the above deﬁnition “3” as the order of smoothness and the order of the critical point is replaced by any other odd number. To ﬁx our ideas, we will always place the critical point of f at 0 ∈ T. 4MICHAELYAMPOLSKY Being a homeomorphism of the circle, a critical circle map f has a well- deﬁned rotation number, denoted ρ( f ). Itisuseful torepresent ρ( f ) as a contined fraction with positive terms (1.1) ρ( f ) = r + r + r + ··· Further on we will abbreviate this expression as [r , r , r , ...] for typographical 0 1 2 convenience. Note that the numbers r are determined uniquely if and only if ρ( f ) is irrational. In this case we shall say that ρ( f ) (or f itself ) is of the type bounded by B if sup r ≤ B; it is of a periodic type if the sequence {r } is periodic. Recall i i k k that an iterate f (0) is called a closest return of the critical point if the arc [0, f (0)] contains no iterates f (0) with i<k. By a classical result of Poincar´eevery circle homeomorphism f with an irrational rotation number is semi-conjugate to the rigid rotation R . Yoccoz [Yoc] has shown that in the case when f is a critical ρ( f ) circle map. This implies, in particular, that if we denote {p /q } the sequence of m m best rational approximations of ρ( f ) obtained as the truncated continued fractions p /q =[r , r ,..., r ], then the iterates { f (0)} are closest returns of 0.Set m m 0 1 m−1 I ≡[0, f (0)]. An important one-parameter family of examples of critical circle maps, the so-called standard (or Arnold’s) maps, is obtained as follows. Given any θ ∈ R the map A (x) = x + θ − sin 2π x 2π commutes with the unit translation: A(x + 1) = A(x) + 1. Therefore it has a well- deﬁned projection to an endomorphism of the circle. We denote this endomor- phism f , although in reality it only depends on the class θ mod Z.Anelementary computation shows that every A is strictly monotone, and has critical points at integer values of x, all of cubic type. Therefore, each f is a critical circle mapping. The considerations of monotone dependence on the parameter imply that θ → ρ(θ) ≡ ρ( f ) is a continuous monotone map of T onto itself. Whenever t ∈ T is irrational, −1 −1 ρ (t) is a single point. For t = p/q the set ρ (t) is a closed interval, for every parameter value in this interval the homeomorphism f has a periodic orbit with a combinatorial rotation number p/q. This orbit has eigenvalue one at the two endpoints of the interval. HYPERBOLICITY OF RENORMALIZATION 5 2. Universality and renormalization 2.1. The golden-mean universality phenomenon The raison d’etre of the renormalization theory of critical circle maps is a uni- versality phenomenon numerically observed as early as in mid-1970’s in one-par- ameter families like the Arnold’s family described above (see [FKS], [ORSS]). For µ ∈ T let f be a one-parameter family of critical circle maps whose dependence on the parameter is smooth and strictly monotone, with ρ( f ) = 0. Let us denote by I (x) the set of parameter values µ with ρ( f ) = x.It isnot difﬁcult to show that I (x) is a single point for irrational values of x;inthe case when x is rational, I (x) is a closed interval. Consider now the intervals I ≡ I ([r , r ,..., r ]) with r = r = ··· = r = 1. The maps with parameter values 0 1 n−1 0 1 n−1 in I possess periodic orbits with periods q ,where p /q is the n-th convergent of n n n n the inﬁnite continued fraction [1, 1, 1, 1, ...]. The ﬁrst numerical observation one makesisthatthere is a particularnumber δ ≈ 2. 83361 such that the length of −n the interval I behaves asymptotically for large n as const · δ . The rate of the geometric decay δ is universal, that is independent on the chosen one-parameter family f . The second observation applies to the limit of the intervals I , i.e. the µ n parameter value µ for which the map f possesses the golden mean rotation 1 µ number 5 − 1 ρ( f ) = =[1, 1, 1, 1, ...]. Denoting by λ the scaling ratio of the length of the closest return arcs ◦q ◦q n n−1 λ = f (c), c c, f (c) , µ µ 1 1 we observe that λ converge to a constant λ ≈ 0. 7760513.The number λ is n ∗ ∗ again independent of the family. Let us remark, that if we consider homeomorphisms of the circle with a non- ﬂat critical point of the order other than three, the universal constants will change. Using the terminology of statistical physics, the order of the critical point speciﬁes the universality class. 2.2. Deﬁnition of renormalization of critical circle maps The strong analogy with universality phenomena in statistical physics and with the already discovered Feigenbaum-Collett-Tresser universality in unimodal maps, led the authors of [FKS] and [ORSS] to explain the existence of the uni- versal constants by introducing a renormalization operator acting on critical circle maps. The deﬁnition is by no means straightforward. We shall arrive at it after 6MICHAELYAMPOLSKY a brief discussion, which will be helpful in understanding the motivation of the resultsin thispaper. The general strategy of deﬁning a renormalization of a given dynamical sys- tem F : X → X is as follows (c.f. [Lyu2]). Select a proper subset Y of X.For apoint x ∈ Y whose forward orbit under F returns to Y denote R (x) ∈ Y the ﬁrst such return. Provided that it is true for every point in Y, this deﬁnes the ﬁrst return map R : Y → Y.Let A : Y → A(Y) be an afﬁne map, such that the size of A(Y), appropriately understood, is the same as the size of the original −1 space X. The rescaled ﬁrst return map A ◦ R ◦ A is the renormalization of F. For the role of the set Y in the case of a critical circle map f : T → T one takes the union of arcs A = I ∪ I for some m>1, where as before I m m m+1 m stands for [0, f (0)]. This choice was made in [ORSS] and [FKS], and has been followed since, although a different choice would have led to the same results. The q q m m+1 ﬁrst return map R : A → A is deﬁned piecewise by f on I and by f m m m m+1 on I .Toview R as a critical circle map we may identify the neighborhoods m m q q q −q m m+1 m+1 m of points f (0) and f (0) by the iterate f . This identiﬁcation transforms the arc A into a C -smooth closed one-dimensional manifold A , R projects to m m m ˜ ˜ ˜ a smooth homeomorphism R : A → A with a critical point at 0.However,the m m m manifold A does not possess a canonical afﬁne structure; the choice of a smooth identiﬁcation φ : A → R/Z gives rise to a plethora of different critical circle maps, all smoothly conjugate. As we can see from the above discussion, the space of critical circle maps is ill-suited to deﬁne renormalization. The authors of [ORSS] and [FKS] circum- vented this difﬁculty by replacing critical circle maps with different objects: Deﬁnition 2.1. —A commuting pair ζ = (η, ξ) consists of two C -smooth orien- tation preserving interval homeomorphisms η : I → η(I ), ξ : I → ξ(I ),where η η ξ ξ (I) I =[0,ξ(0)], I =[η(0), 0]; η ξ (II) Both η and ξ have homeomorphic extensions to interval neighborhoods of their re- spective domains with the same degree of smoothness, which commute, η ◦ ξ = ξ ◦ η; (III) ξ ◦ η(0) ∈ I ; (IV) η (x) = 0 = ξ (y),for all x ∈ I \{0},and all y ∈ I \{0}. η ξ The commutation condition allows one to iterate the extensions of the maps of a commuting pair. We can also use it to repeat the glueing construction. That is, given a critical commuting pair ζ = (η, ξ) we can regard the interval I = [η(0), ξ ◦ η(0)] as a circle, identifying η(0) and ξ ◦ η(0) and deﬁne f : I → I by η ◦ ξ(x) for x ∈[η(0), 0] f = . η(x) for x ∈[0,ξ ◦ η(0)] HYPERBOLICITY OF RENORMALIZATION 7 FIG. 1. — Acommuting pair The mapping ξ extends to a C -diffeomorphism of open neighborhoods of η(0) and ξ ◦ η(0). Using it as a local chart we turn the interval I into a closed one- dimensional manifold M. Condition (II) above implies that the mapping f projects to a well-deﬁned C -smooth homeomorphism F : M → M. Identifying M with the circle by a diffeomorphism φ : M → T we recover a critical circle mapping φ −1 f = φ◦F ◦φ . The critical circle mappings corresponding to two different choices of φ are conjugated by a diffeomorphism, and thus we recovered a C -smooth conjugacy class of circle mappings from a critical commuting pair. We can metrize the space of C -smooth commuting pairs considered modulo an afﬁne conjugacy as follows (see [dFdM1]). Let ζ = (η ,ξ ), ζ = (η ,ξ ) be 1 1 1 2 2 2 two such pairs, and denote w : C → C aM¨ obius transformation which maps the ordered triple of points η (0), 0, ξ (0) to 0, 1/2, 1.The C -distance between ζ i i 1 and ζ is set to be dist r (ζ ,ζ ) = max |ξ (0)/η (0) − ξ (0)/η (0)|, C 1 2 1 1 2 2 −1 −1 dist r w ◦ ζ ◦ w , w ◦ ζ ◦ w . C 1 1 2 2 1 2 Let f be a critical circle mapping, whose rotation number ρ has a continued fraction expansion (1.1) with at least m+ 1 terms, and let p /q =[r , ..., r ].The m m 0 m−1 q q m+1 m pair of iterates f and f restricted to the circle arcs I and I correspond- m m+1 ingly can be viewed as a critical commuting pair in the following way. Let f be ¯ ¯ the lift of f to the real line satisfying f (0) = 0,and 0< f (0) <1.For each m> 0 let I ⊂ R denote the closed interval adjacent to zero which projects down to the interval I .Let τ : R → R denote the translation x → x + 1.Let η : I → R, m m −p q −p q m+1 m+1 m m ¯ ¯ ¯ ξ : I → R be given by η ≡ τ ◦ f , ξ ≡ τ ◦ f . Then the pair of maps m+1 q q m+1 m ¯ ¯ (η|I ,ξ|I ) forms a critical commuting pair corresponding to ( f |I , f |I ). m m+1 m m+1 Henceforth we shall simply denote this commuting pair by q q m+1 m (2.1) ( f |I , f |I ). m m+1 This allows us to readily identify the dynamics of the above commuting pair with that of the underlying circle map, at the cost of a minor abuse of notation. 8MICHAELYAMPOLSKY Following [dFdM1], we say that the height χ(ζ) of a critical commuting pair ζ = (η, ξ) is equal to r,if r r+1 0 ∈[η (ξ(0)), η (ξ(0))]. If no such r exists, we set χ(ζ) =∞, in this case the map η|I has a ﬁxed point. For a pair ζ with χ(ζ) = r< ∞ one veriﬁes directly that the mappings r r η|[0,η (ξ(0))] and η ◦ ξ|I again form a commuting pair. For a commuting pair ζ = (η, ξ) we will denote by ζ the pair ( η|I , ξ |I ) where tilde means rescaling by η ξ the linear factor λ =− . |I | Deﬁnition 2.2. —The renormalization of a real commuting pair ζ = (η, ξ) is the commuting pair r r Rζ = (η ◦ ξ |I , η|[0,η (ξ(0))]). r r The non-rescaled pair (η ◦ ξ|I ,η|[0,η (ξ(0))]) will be referred to as the pre- renormalization pRζ of the commuting pair ζ = (η, ξ). For a pair ζ we deﬁne its rotation number ρ(ζ) ∈[0, 1] to be equal to the continued fraction [r , r , ...] where r = χ(R ζ). In this deﬁnition 1/∞ is under- 0 1 i stood as 0, hence a rotation number is rational if and only if only ﬁnitely many renormalizations of ζ are deﬁned; if χ(ζ) =∞, ρ(ζ) = 0. Thus deﬁned, the ro- tation number of a commuting pair can be viewed as a rotation number in the usual sense: Proposition 2.1. — The rotation number of the mapping F is equal to ρ(ζ). There is an advantage in deﬁning ρ(ζ) using a sequence of heights in removing the ambiguity in prescribing a continued fraction expansion to rational rotation numbers in a renormalization-natural way. For ρ =[r , r , ...]∈[0, 1] letusset 0 1 G(ρ) =[r , r ,...]= , 1 2 where {x} denotes the fractional part of a real number x (G is usually referred to as the Gauss map). As follows from the deﬁnition, ρ(Rζ) = G(ρ(ζ)) for a real commuting pair ζ with ρ(ζ) = 0. The renormalization of the real commuting pair (2.1), associated to some q q m+2 m+1 critical circle map f ,is the rescaled pair ( f |I , f |I ).Thusfor a given m+1 m+2 critical circle map f the renormalization operator recovers the (rescaled) sequence of the ﬁrst return maps: q q i+1 ( f |I , f |I ) . i i+1 i=1 HYPERBOLICITY OF RENORMALIZATION 9 2.3. Explanation of universality: Lanford’s Program The golden-mean universality phenomenon described above was explained in [ORSS] and [FKS] by a conjectural hyperbolicity property of the operator R. The conjecture was later generalized by various authors, particularly by Lanford [Lan2], to account for more complex universalities. We present it below in its most general form, known as Lanford’s Program. First, a deﬁnition. A critical commuting pair is a commuting pair (η, ξ) whose maps are real-analytic. We shall also impose a technical assumption that ξ analyt- ically extends to an interval (a, b) 0 with ξ(a, b) ⊃[η(0), ξ(0)], and has a single critical point 0 in this interval. The space of critical commuting pairs modulo afﬁne conjugacy will be denoted by C; its subset consisting of pairs ζ with χ(ζ) =∞ will be denoted by S . Renormalization is a transformation R : C \C → C. It ∞ ∞ is not difﬁcult to show (see [Ya2]) that this transformation is injective: Proposition 2.2. —The map R : C \S → C is one-to-one. To outline Lanford’s Program, let us give an informal description of the action of renormalization on the space of critical commuting pairs. −1 Let ρ = φ(θ) : T → T be a monotone continuous function such that φ (ρ) is a point for ρ irrational and an interval otherwise. An example of such a function is the dependence θ → ρ( f ) of the rotation number of a standard map on the parameter. Imagine the space of critical commuting pairs as an inﬁnite-dimensional cylinder C = T ×C , where the rotation number of a commuting pair ζ(θ, ·) with the equatorial coordinate θ ∈ T is ρ(θ). The cylinder C is partitioned into strips (cf. Figure 2) S ={ζ ∈ C|ρ(ζ) =[r, r , ...]} for r = 1, ..., ∞ . r 1 FIG. 2. 10 MICHAEL YAMPOLSKY A boundary component of the strip S is the hypersurface P ⊂ S with the ∞ ∞ ∞ property that a pair ζ ∈ P if and only if it has a single ﬁxed point with unit eigenvalue (ζ is a “parabolic pair”). The sets S accumulate on P in a clockwise r ∞ direction. It is natural to think of the transformation R : C \S → C as being deﬁned piecewise, given on each S , r =∞ by the formula: R : (η, ξ) → (η, η ◦ ξ). The operator R expands the strip S in the equatorial direction, mapping it onto r r a thin equatorial cylinder, which intersects all the vertical strips. We may close the domain of R by adjoining the hypersurface P to it and setting R on P to ∞ ∞ an appropriately deﬁned (multivalued) parabolic renormalization transformation P, with this convention the image of P is also an equatorial cylinder. Lanford con- jectured that this picture can be appropriately endowed by a smooth metric, so that the operator R is uniformly expanding in the θ direction and uniformly con- tracting in the complementary directions. Thus R posesses an inﬁnite-dimensional “horseshoe” structure. The “boxes” R (S ∩ R (S ∩ ··· (R S ) ··· ) r r r r r r −1 −1 −2 −2 −n −n −1 −1 −1 ∩S ∩ R R ··· R S ··· r r 0 n r r r 0 1 n−1 are exponentially small in n, and their intersection is the R-invariant hyperbolic Cantor set A parametrized by the bi-inﬁnite sequences of r ∈ N ∪{∞}.The set A is a “horseshoe” attractor for the renormalization operator, dist(R ζ, A ) → 0 at a geometric rate. The action of R on A is conjugate to the two-sided shift. Let us conclude this exposition by formalizing the Hyperbolicity Conjecture as follows: Renormalization Hyperbolicity Conjecture (Lanford’s Program). — The renormalization operator R in the space of critical commuting pairs possesses a “horseshoe” attractor A on which its action is conjugated to the two-sided shift. Moreover, there exists a renormalization- invariant space of critical commuting pairs with the structure of an inﬁnite dimensional smooth manifold, with respect to which A is a hyperbolic set with one-dimensional expanding direction. If t denotes the expanded coordinate in a local chart, then the dependence t → ρ(t, ·) is continuous and (not strictly) monotone. It is not difﬁcult to see that this Conjecture explains, in particular, the golden mean universality phenomenon we described. Indeed, it implies the existence of a unique ﬁxed point ζ = (η ,ξ ) ∈ A with ρ(ζ ) = ( 5 − 1)/2.The ﬁxed point ∗ ∗ ∗ ∗ ζ is hyperbolic, let us denote its unique expanding eigenvalue δ and set the ∗ ∗ HYPERBOLICITY OF RENORMALIZATION 11 scaling factor |I |/|I |= λ .Then for any ζ with ρ(ζ) = ( 5 − 1)/2 we have ξ η ∗ ∗ ∗ ◦n R ζ → ζ which explains the universality of λ .Moreover, let ζ be a mono- ∗ ∗ µ −n tone one-parameter family of critical commuting pairs. Since the slices (R ) (S ) 1 1 accumulate at the local stable manifold of ζ at the geometric rate 1/δ ,they ∗ ∗ −n have “width” ∼ δ .Providedthe family ζ crosses the local stable manifold of ζ µ ∗ transversely, this implies that the sizes of parameter intervals I ={µ : ζ ∈ S } n µ n as well. decay at the rate 1/δ 3. The passage from smooth to holomorphic The golden-mean universality is observed in any one-parameter family of smooth critical circle maps. We chose to formulate the Hyperbolicity Conjecture for analytic commuting pairs, in part to ensure that the action of renormalization is injective. A deeper reason is the following. In 1986 Eckmann and Epstein [EE] introduced a space of critical commuting pairs now known as the Epstein class. They showed that this class is invariant under the action of R, and constructed the golden-mean ﬁxed point of R in this class using the methods of geometric complex analysis. It was further shown by various people, such as Sullivan (in the unimodal case), Swia¸tek, Herman, and Yoccoz, that the renormalizations of any C -smooth commuting pair with an irrational rotation number converge to the Epstein class, at a geometric rate in the C -metric. Below, after some preliminaries, we deﬁne the Epstein class, and formulate these results more precisely. 3.1. Carathe´odory topology on a space of branched coverings Consider the collection X of all triplets (U, u, f ),where U ⊂ C is a topo- logical disk different from the whole plane, u ∈ U,and f : U → C is a three-fold analytic branched covering map, with the only branch point at u. We will topol- ogize X as follows (cf. [McM1]). Let {(U , u )} be a sequence of open connected regions U ⊂ C with marked n n n points u ∈ U . Recall that this sequence Caratheodory converges to a marked region n n (U, u) if: • u → u ∈ U,and • for any Hausdorff limit point K of the sequence C \ U , U is a component of C \ K. For a simply connected U ⊂ C and u ∈ U let R : D → U denote the inverse (U,u) Riemann mapping with normalization R (0) = u, R (0) >0. By a classical (U,u) (U,u) result of Caratheodory, the Carathedory convergence of simply-connected regions ´ ´ 12 MICHAEL YAMPOLSKY (U , u ) → (U, u) is equivalent to the locally uniform convergence of the inverse n n Riemann mappings R to R . (U ,u ) (U,u) n n For positive numbers , , and compact subsets K and K of the open 1 2 3 1 2 unit disk D, let the neighborhood U (U, u, f ) of an element (U, u, f ) ∈ X , , ,K ,K 1 2 3 1 2 be the set of all (V, v, g) ∈ X,for which: •|u − v| < , • sup |R (z) − R (z)| < , (V,v) (U,u) 2 z∈K • and R (K ) ⊂ V,and sup |f (z) − g(z)| < . (U,u) 2 3 z∈R (K ) (U,u) 2 One veriﬁes that the sets U (U, u, f ) form a base of a topology on X, , , ,K ,K 1 2 3 1 2 which we will call Caratheodory topology. This topology is clearly Hausdorff, and the convergence of a sequence (U , u , f ) to (U, u, f ) is equivalent to the Caratheodory n n n convergence of the marked regions (U , u ) → (U, u) as well as a locally uniform n n convergence f → f . 3.2. The Epstein class An orientation preserving interval homeomorphism g : I =[0, a]→ g(I) = J belongs to the Epstein class E if it extends to an analytic three-fold branched cov- ering map of a topological disk G ⊃ I onto the double-slit plane C ,where J ⊃ cl J. Any map g in the Epstein class can be decomposed as (3.1) g = Q ◦ h, where Qc(z) = z + c,and h : I →[0, b] is a univalent map h : G → ∆ (h) onto the complex plane with six slits, which triple covers C under the cubic map Qc(z). For any s ∈ (0, 1), let us introduce a smaller class E ⊂ E of Epstein map- −1 pings g : I =[0, a]→ J ⊂ J for which both |I| and dist(I, J) are s -commensurable with | J|, the length of each component of J \ J is at least s| J|,and g (a) >s.We will often refer to the space E as the Epstein class, and to each E as an Epstein class. C belongs to the (an) Epstein class if We say that a commuting pair (η, ξ) ∈ both of its maps do. Similarly, a critical circle map f is Epstein if Rf is in the Epstein class. It immediately follows from the deﬁnitions that: Lemma 3.1. — If a renormalizable commuting pair ζ is in the Epstein class, then the same is true for Rζ . Let us make a note of an important compactness property of E s HYPERBOLICITY OF RENORMALIZATION 13 Lemma 3.2 (Lemma 2.10 [Ya2]). — Let s ∈ (0, 1). The collection of normalized maps g ∈ E with I =[0, 1], with marked domains (U, 0) is sequentially compact with respect to Caratheodory topology. 3.3. Real a priori bounds An excellent exposition of the results of Herman, Swia¸tek, and Yoccoz on the real geometry of critical circle maps is found in [dFdM1]. We formulate some of the main results. Real a priori bounds of Herman and Swia¸tek. — There exists a universal constant K such that the following holds. Let f be a critical circle map. Denote Π the partition of the circle by the iterates q −1 q −1 q −1 q −1 q i m−1 q i m q +q i m−1 q +q i m m m−1 m−1 m m m−1 { f (0)} ∪{ f (0)} ∪{ f (0)} ∪{ f (0)} i=0 i=0 i=0 i=0 (note that I and I are two adjacent intervals of the partition). Then there exists M = m m−1 M( f ) such that for every m>M any two adjacent intervals of the partition are K- 1 m ∞ commensurable. Moreover, the C -norm of the renormalizations {R ( f )} is bounded m=M by K. Utilizing the above bounds Herman has shown in [He] (see [dFdM1] for a published account): Theorem 3.3. — Any two critical circle maps with the same irrational rotation numbers are quasisymmetrically conjugate. The importance of the Epstein class lies in the fact that all C -limit points m ∞ of the sequence {R ( f )} are in E for a universal value of s> 0.A more m=M precise formulation of this was proved in the recent work of de Faria and de Melo [dFdM1]: Lemma 3.4. — There exists a universal constant s> 0 such that the following holds. Let f ∈ C , (r ≥ 3) be a critical circle map with an irrational rotation number. Then the m r−1 q q m+1 sequence of real commuting pairs R ( f ) = ( f |I , f |I ) is bounded in C -metric, m m+1 r−1 and C -converges to E at a geometric rate. In particular, for a critical circle map f ∈ E there exists σ >0 such that all its renormalizations are contained in E . Moreover, the constant σ can be chosen independent on f , after skipping the ﬁrst few renormalizations. Finally, let us formulate a technical statement about Epstein pairs to be used further in the paper: 14 MICHAEL YAMPOLSKY Lemma 3.5 (Lemma 2.13, [Ya2]). — Let ζ = (η, ξ) ∈ E be a critical commuting pair with ρ(ζ) = 0, which appears as a limit of a sequence {ζ }⊂ E with ρ(ζ ) ∈ R \ Q . n n Then the map η has a unique ﬁxed point in the interval I , which is necessarily parabolic, with multiplier one. Acommuting pair ζ = (η, ξ) ∈ E will be called parabolic if the map η has a unique ﬁxed point in I , which has a unit multiplier; this point will usually be denoted p . Note, that by virtue of its uniqueness, p has to be globally attracting η η on one side for the interval homeomorphism η| , it is globally attracting on the −1 other side under η . 4. Outline of the results 4.1. Known results The early efforts to prove the Renormalization Hyperbolicity Conjecture cul- minated in an argument of Mestel [Mes], who established the existence and local hyperbolicity properties of a ﬁxed point of R in C . The argument was based on rigorous computer estimates. This approach was developed by Lanford in the context of the renormalization of unimodal maps. One of its major drawbacks is its local nature; neither the uniqueness of the ﬁxed point, nor global hyperbolicity of R are established. In his address to ICM-86 in Berkeley Sullivan [Sul1] outlined a program of constructing the invariant set of renormalization of unimodal maps and its stable set by means of the Teichmul¨ ler theory, which he subsequently carried out (see [Sul2],[MvS]). Sullivan’s program has been adapted to the setting of critical circle mappings by de Faria [dF1],[dF2]. De Faria has deﬁned, in particular, extensions of Epstein commuting pairs to holomorphic dynamical systems in the plane, which are analogous to quadratic-like mappings in the unimodal renormalization theory. We discuss these extensions, called holomorphic commuting pairs,in §5. Using Sullivan’s Riemann surface laminations machinery, de Faria constructed a horseshoe attractor A for the renormalization operator acting on the space of Epstein commuting pairs of type bounded by some constant B (this corresponds to restricting the operator R to the ﬁnite collection of vertical strips ∪S , r ≤ B in Figure 2). In a recent paper [Ya2] we generalized the results of de Faria to Epstein commuting pairs of unbounded combinatorial type. Denote Σ the space of bi-inﬁnite sequences (..., r , ..., r , r , r , ..., r , ...) with r ∈ N ∪{∞} equipped with the weak topology. −k −1 0 1 k i Our theorem completely settles the ﬁrst part of Lanford’s Program: Theorem 4.1 (Renormalization horseshoe). — There exists an R-invariant closed set A ⊂ E consisting of pairs with irrational rotation numbers and parabolic pairs with the HYPERBOLICITY OF RENORMALIZATION 15 following properties. The action of R on A is topologically conjugate to the two-sided shift ¯ ¯ σ : Σ → Σ: −1 i ◦ R ◦ i = σ −1 and if ζ = i (..., r , ..., r , r , r , ..., r , ...) then ρ(ζ) =[r , r , ..., r , ...].For any −k −1 0 1 k 0 1 k ζ ∈ E with irrational rotation number we have R ζ → A in the Caratheodory topology. Moreover, for any two pairs ζ , ζ ∈ E with ρ(ζ) = ρ(ζ ) we have n n dist(R ζ, R ζ ) → 0 for the uniform distance between analytic extensions of the renormalized pairs on compact sets. The proof of Theorem 4.1 combines the ideas of McMullen from [McM2] with apriori bounds obtained in [Ya1]. We consider the geometric limits of the rescaled sequences of renormalizations ζ, Rζ, ..., R ζ. These limits which we call bi-inﬁnite limiting renormalization towers (see §5for the deﬁnition) should be thought of as bi-inﬁnite sequences of nested holomorphic commuting pairs in the plane. Each tower corresponds to a single point in the attractor A , and the result follows from a Rigidity Theorem [Ya2]: Theorem 4.2 (Tower rigidity). — Any two towers with the same rotation numbers are afﬁnely conjugated. As far as the hyperbolicity aspects of Lanford’s Program, the following is known. In the case of a bounded type rotation number, de Faria and de Melo [dFdM2] strengthened the original result of de Faria, utilizing a technique of Mc- Mullen [McM2] to show that the rate of converegence to A is geometric: Theorem 4.3 ([dFdM2]). — For every B ∈ N and every r ∈ N ∪ 0 there exists α <1 such that for every ζ ∈ E with ρ(ζ) ∈ R \ Q of type bounded by B and for every r,B ˆ ˆ ζ ∈ A with ρ(ζ) = ρ(ζ) we have n n n dist (R ζ, R ζ) < const ·(α ) . C r,B Despite the difﬁculty in imposing a manifold structure on the space of com- muting pairs compatible with the analyticity of R, it has long been known how one may construct an unstable direction in such a manifold. A version of this construction was ﬁrst shown to us by Folkert Tangerman, who participated in its discussion in Sullivan’s seminar at the Graduate Center of CUNY. We give a rather simple argument in §9. 16 MICHAEL YAMPOLSKY 4.2. Statement of the main result First, some deﬁnitions. Suppose, B is a complex Banach space whose elem- ents are functions of the complex variable. Let us say that the real slice of B is the real Banach space B consisting of the real-symmetric elements of B.If X is a Banach manifold modelled on B with the atlas {Ψ } we shall say that X is −1 R R real-symmetric if Ψ ◦Ψ (B ) ⊂ B for any pair of indices γ , γ .The real slice of X γ 1 2 1 γ R −1 R is then deﬁned as the real Banach manifold X ⊂ X given by Ψ (B ) in a local R R chart Ψ .Anoperator A deﬁned on a subset of X is real-symmetric if A(X ) ⊂ X . A particular example we shall introduce in §7 is a real-analytic manifold C of critical cylinder maps. Its real slice M ≡ C consists of critical circle maps deﬁned in an annulus U ⊃ T in the cylinder C/Z. We will construct a closed equivalence relation denoted ∼ between com- conf muting pairs in the Epstein class E (with s being as in Lemma 3.4) with the property that if ζ ∼ ζ , then the analytic extensions of the commuting pairs are 1 conf 2 conjugate in a speciﬁc neighborhood of the interval of deﬁnition by a conformal change of coordinates. In particular, ρ(ζ ) = ρ(ζ ). Moreover, there exists a value 1 2 N N N ∈ N such that if ζ and ζ are N-times renormalizable, then R ζ ∼ R ζ . 1 2 1 conf 2 This last property implies that periodic orbits of R constructed in Theorem 4.1 project to the quotient space. The main result of this paper is the Renormalization Hyperbolicity Theorem in the following form: Theorem 4.4 (Renormalization Hyperbolicity). — There exists a real-symmetric analytic operator R from an open subset of C to C (thus R maps an open subset of M cyl U U cyl into M), such that the following holds. For every r =∞ there is a canonical homeomorphism onto the image S ∩ E / ∼ −→ M r s conf N −1 such that ρ(ι (ζ)) = ρ(R(ζ)),and ι ◦ R ◦ ι ≡ R . For every periodic point p ∈ S r r cyl r of the operator R there exists ∈ N such that the point ι ( p) is a hyperbolic ﬁxed point of R | , with a one-dimensional unstable direction. cyl Note, in particular, that we have found a way to deﬁne a renormalization for critical circle maps themselves, without recourse to critical commuting pairs. Our theorem implies the existence of universal constants associated to every rotation number of a periodic type, and in particular we get a version of the golden-mean universality: Corollary 4.5 (Golden-Mean Universality). — There exists a pair of universal constants δ >1 and λ >0, such that the following holds. Let { f } be a one-parameter family of analytic µ HYPERBOLICITY OF RENORMALIZATION 17 critical circle maps with a C -smooth dependence on the parameter µ.Denote f : R → R a continuously chosen lift of f : T → T, and assume additionally that ∂f (x)/∂µ >0 for µ µ any x ∈ R (the Arnold’s family is one example of such family). Suppose ρ( f ) = ( 5−1)/2 and denote I the closed interval of values of the parameter for which ρ( f ) = p /q where n µ n n p /q is the n-th convergent of the golden mean. Then: n n lM •|I |/δ → a> 0 for some M ∈ N; k+lM q q n n+1 •| f (0)|/| f (0)|→ λ geometrically fast. µ µ ∗ ∗ We remark that the second statement already follows from [dFdM2]. 5. Holomorphic commuting pairs 5.1. Deﬁnitions De Faria [dF1,dF2] introduced holomorphic commuting pairs to apply Sul- livan’s Riemann surface laminations technique to the renormalization of critical circle maps. They are suitably deﬁned holomorphic extensions of critical commut- ing pairs which replace Douady-Hubbard polynomial-like maps [DH2]. A critical commuting pair ζ = (η| ,ξ| ) extends to a holomorphic commuting pair H if there I I η ξ exist four simply-connected R-symmetric domains ∆ , D, U, V such that ¯ ¯ ¯ ¯ ¯ • D, U, V ⊂ ∆ , U ∩ V ={0}; the sets U \ D, V \ D, D \ U,and D \ V are nonempty, connected, and simply-connected, U ∩ R = I , V ∩ R = I ; η ξ • mappings η : U → (∆ \ R) ∪ η(I ) and ξ : V → (∆ \ R) ∪ ξ(I ) are onto η ξ and univalent; • ν ≡ η ◦ ξ : D → (∆ \ R) ∪ ν(I ) is a three-fold branched covering with a unique critical point at zero, where I = D ∩ R. We shall call ζ the commuting pair underlying H , and write ζ ≡ ζ .The domain D ∪ U ∪ V of a holomorphic commuting pair H will be denoted Ω or Ω ,the FIG. 3. — A holomorphic commuting pair 18 MICHAEL YAMPOLSKY rangewill bedenoted ∆ or ∆ . The closure of the set of points whose orbit under H is contained in Ω will be referred to as the ﬁlled Julia set of H , denoted K(H ).The Julia set of H is deﬁned as J(H ) = ∂K(H ). It is easy to see directly from the deﬁnition (cf. [dF2]) that: Proposition 5.1. —Let ζ be a commuting pair with χ(ζ) < ∞. Suppose ζ is a re- striction of a holomorphic commuting pair H , that is ζ = ζ . Then there exists a holomorphic commuting pair G with range ∆ , such that ζ = Rζ . H G The shadow of the holomorphic commuting pair H is the following piecewise- deﬁned holomorphic dynamical system: η(z), z ∈ U S (z) = ξ(z), z ∈ V ξ ◦ η(z), z ∈ D \ (U ∪ V). As the next proposition shows one may think of the shadow of a holomorphic commuting pair as an analogue of a cubic-like map: Proposition 5.2 (Prop. II.4. [dF2]). — Given a holomorphic commuting pair H as −1 above, consider its shadow S .Let I = Ω ∩ R,and X = I ∪ S (I).Then: • The restriction of S to Ω \ X is a regular three fold covering onto ∆ \ R. • S and H share the same orbits as sets. We will say that two holomorphic commuting pairs H : Ω → ∆ and H H G : Ω → ∆ are conjugate if there is a homeomorphism h : ∆ → ∆ such G G G H that −1 S = h ◦ S ◦ h. G H −1 In this case we will simply write G = h ◦ H ◦ h. 5.2. Complex a priori bounds We shall denote by H the space of holomorphic commuting pairs H : Ω → ∆ whose underlying real commuting pair (η, ξ) is in the Epstein class. In this case both maps η and ξ extend to triple branched coverings η ˆ : U → ∆ ∩ C η( J ) ˆ ˆ and ξ : V → ∆ ∩ C respectively. We will turn H into a topological space by ξ( J ) ˆ ˆ ˆ identifying it with a subset of X ×X by H → (U, 0, η) × (V, 0, ξ) (cf. §3.1). We say that a real commuting pair (η, ξ) with an irrational rotation num- ber has complex apriori bounds, if all its renormalizations extend to holomorphic commuting pairs with bounded moduli: mod(∆ \ Ω ) > µ >0. HYPERBOLICITY OF RENORMALIZATION 19 For µ ∈ (0, 1) let H(µ) denote the space of holomorphic commuting pairs H : Ω → ∆ ,with mod(∆ \ Ω ) > µ, min(|I |, |I |) > µ and diam(∆ ) <1/µ. H H H H η ξ H Lemma 5.3 (Lemma 2.15 [Ya2]). — For each µ ∈ (0, 1) the space H(µ) is sequen- tially pre-compact, with every limit point contained in H(µ/2). The existense of complex apriori bounds is a key analytic issue of renor- malization theory. In the case of critical circle maps it is settled by the following theorem: Theorem 5.4. — There exist universal constants µ >0 and K> 1 such that the following holds. Let ζ ∈ C be a critical commuting pair with an irrational rotation number. Then there exists N = N(ζ) such that for all n ≥ N the commuting pair R ζ extends to a holomorphic commuting pair H : Ω → ∆ in H(µ).The range ∆ is a Euclidean disk, n n n n and the regions Ω ∩ (±H) are K-quasidisks. Remark 5.1. — We ﬁrst proved this theorem in [Ya1] for commuting pairs ζ in an Epstein class E ,inwhich case N = N(s). Our proof was later adapted by de Faria and de Melo [dFdM2] to the case of a non-Epstein critical commuting pair. In the general case, in a Carath´ eodory compact family of critical commuting pairs, the number N canbechosenuniformly. Let ζ be at least n times renormalizable critical commuting pair. For the lack of a better term, let us say that the pair of numbers τ (ζ) = (r , r ) forms n n−1 n−2 the history of the pair R ζ . Based on the above theorem and a detailed analysis of the shapes of the domains Ω we proved the following in [Ya1]: Theorem 5.5 ([Ya1]). — There exists a universal constant K >1 such that the follow- ing holds. Let ζ = (η ,ξ ) and ζ = (η ,ξ ) be two critical commuting pairs with irrational 1 1 1 2 2 2 rotation numbers. Let n> max(N(ζ ), N(ζ )) + 1 as above. Assume that the n-th renormal- 1 2 izations of ζ , ζ have the same rotation number and the same history. Then their holomorphic 1 2 1 2 commuting pair extensions H , H are K −quasiconformally conjugate. The conjugating map n n is conformal on the ﬁlled Julia set. For commuting pairs of the type bounded by B this theorem was ﬁrst proved by de Faria [dF1,dF2], with “K ” depending on the value of B. 5.3. Limiting renormalization towers Sullivan’s original proof of the existence and uniqueness of the Feigenbaum ﬁxed point used the Teichmuller theory for Riemann surface laminations asso- ciated to quadratic-like maps, which he himself developed. McMullen [McM2] 20 MICHAEL YAMPOLSKY has replaced this approach with an elegant argument based on a rigidity result for dynamical objects he called renormalization towers. The renormalization tow- ers of critical commuting pairs were studied in [dFdM2] for pairs of bounded type, and later in [Ya2] without the restriction on the type. Their deﬁnition is as follows. Let {H } be a sequence of generalized holomorphic commuting pairs. Set ζ = ζ . Suppose m(k) >n(k) are two sequences of natural numbers such k H that n(k) →∞, m(k) − n(k) →∞ and ζ is m(k)-times renormalizable. Moreover, n(k)+i i the pre-renormalizations pR ζ have holomorphic pair extensions H such that i ∞ H → H ∈ H,for i ∈ Z. Then the bi-inﬁnite sequence T = (H ) is referred i i k −∞ to as a bi-inﬁnite limiting renormalization tower.The rotation number ρ(T ) is naturally deﬁned as the bi-inﬁnite sequence of heights (χ(ζ )) . The precise formulation i −∞ of Theorem 4.2 is: Tower Rigidity Theorem ([Ya2]). — For any bi-inﬁnite sequence ρ ¯ = (χ ) ,χ ∈ n n −∞ N∪{∞} there exists a bi-inﬁnite limiting renormalization tower T with ρ(T ) = ρ ¯.Moreover, 1 ∞ 2 ∞ for any two limiting renormalization towers T = (H ) , T = (H ) with ρ(T ) = 1 2 1 i −∞ i −∞ ρ(T ) we have ζ 1 ≡ ζ 2 for all i. H H i i 6. Parabolic mapsand theirperturbations General facts. — We begin with a brief review of the theory of parabolic bifurcations, as applied in particular to an interval map in the Epstein class. For a more comprehensive exposition the reader is referred to [Do], supporting tech- nical details may be found in [Sh]. Fix a map η ∈ E having a parabolic ﬁxed point p with unit multiplier. A R Theorem 6.1 (Fatou Coordinates). — There exist topological discs U and U , called attracting and repelling petals, whose union is a punctured neighborhood of the parabolic periodic point p such that A A k A ¯ ¯ η (U ) ⊂ U { p}, and η (U ) ={ p}, k=0 R R −k R ¯ ¯ η (U ) ⊂ U { p}, and η (U ) ={ p}, k=0 −1 where η is the univalent branch ﬁxing ζ . Moreover, there exist injective analytic maps A A R R Φ : U → C and Φ : U → C, HYPERBOLICITY OF RENORMALIZATION 21 unique up to post-composition by translations, such that A A R R Φ (η (z)) = Φ (z) + 1 and Φ (η (z)) = Φ (z) − 1. 0 0 A A R R The Riemann surfaces C = U /η and C = U /η are conformally equivalent to the 0 0 cylinder C/Z. Proof. — To ﬁx our ideas, let us assume that p>0. We will show that p is a simple parabolic point of η ,thatis, 2 2 η (z − p) = (z − p) + a(z − p) + o((z − p) ), with a = 0. Indeed, for a small c<0 consider the map η = η +c.As η ∈ E,there c 0 c exists a well-deﬁned inverse branch φ of this map in C \ R which preserves both the upper and the lower half-planes. Since η has no real ﬁxed points, the Denjoy- Wolff Theorem implies that there exists a pair of complex conjugate ﬁxed points ± + p ∈±H such that every orbit originating in H converges to p and similarly c c for the other ﬁxed point. Thus, p splits into a pair of repelling ﬁxed points, and hence its algebraic multiplicity is two. The rest of the construction is classical. A A R R We denote π : U → C and π : U → C the natural projections. The A R A R quotients C and C are customarily referred to as Ecalle-Voronin cylinders;we will ﬁnd it useful to regard these as Riemann spheres with distinguished points +, − A A ﬁlling in the punctures. The real axis projects to the natural equators E ⊂ C and R R A R E ⊂ C .Any conformal transit homeomorphism τ : C → C ﬁxing the ends +, − is a translation in suitable coordinates. Liﬁting it produces a map τ : U → C satisfying τ ◦ π = π ◦ τ. A R We will sometimes write τ ≡ τ ,and τ ¯ = τ ¯ ,where θ θ R A −1 Φ ◦ τ ◦ (Φ ) (z) ≡ z + θ mod Z. Suppose for an Epstein map η in a sufﬁciently small neighborhood of η the parabolic point splits into a complex conjugate pair of repelling ﬁxed points p ∈ H ± 2π i±α(η) and p ¯ with multipliers λ = e . In this situation one may still speak of attracting and repelling petals: Lemma 6.2 (Douady Coordinates). — There exists a neighborhood U(η ) ⊂ E of the map η such that for any η ∈ U(η ) with | arg α(η)| < π/4, there exist topological discs 0 0 A R U and U whose union is a neighborhood of p, and injective analytic maps η η A A R R Φ : U → C and Φ : U → C η η f 22 MICHAEL YAMPOLSKY unique up to post-composition by translations, such that A A R R Φ (η(z)) = Φ (z) + 1 and Φ (η(z)) = Φ (z) + 1. η η η η A A R R The quotients C = U /η and C = U /η are Riemann surfaces conformally equivalent to η η η η C/Z. Let us note: Proposition 6.3. — There exists an open neighborhood W (η ) ⊂ E of η in the 0 s 0 Caratheodory topology, such that for every η ∈ W (η ) as above, the condition on the eigenvalues of the repelling ﬁxed points is automatically satisﬁed. Proof. — Recall that if z is an isolated ﬁxed point of an analytic map f , then the holomorphic index of z is the residue 1 dz i( f , z ) = 2π i z − f (z) where the integrating is done over a small contour enclosing z (see [Mil]). In the case when λ = f (z) = 1,anelementary computationshows that i( f , z ) = 1/(1 − λ).In our case,bycontinuity, 1 1 + −→ i(η , p). + − η→η 1 − λ 1 − λ 0 η η + ± Since the right-hand side is ﬁnite, and λ = λ ,and 1 − λ −→ 0, it follows that η η η η→η arg 1 − λ −→ . η→η 0 2 A R An arbitrary choice of real basepoints a ∈ U and r ∈ U enables us to specify the Fatou and Douady coordinates uniquely, by requiring that Φ (a) = A R R Φ (a) = 0,and Φ (r) = Φ (r) = 0. The following fundamental theorem ﬁrst η η appeared in [DH1]: A R Theorem 6.4. — With these normalizations the maps Φ , Φ depend continuously on η η η with respect to the compact-open topology, and A A R R Φ → Φ and Φ → Φ η η A R uniformly on compact subsets of U and U respectively. n(η) Moreover, select the smallest n(η) ∈ N for which η (a) ≥ r.Then −1 n(η) R A η (z) = Φ ◦ T ◦ Φ θ(η)+K η η HYPERBOLICITY OF RENORMALIZATION 23 wherever both sides are deﬁned. In this formula T (z) denotes the translation z → z + a, θ(η) ∈[0, 1) is given by θ(η) = 1/α(η) + o(1) mod 1, α(η)→∞ and the real constant K is determined by the choice of the basepoints a, r. Thus for a sequence n(η ) {η }⊂ U(η) converging to η, the iterates η converge locally uniformly if and only if there is a convergence θ(η) → θ , and the limit in this case is a certain lift of the transit homeomorphism τ for the parabolic map η . θ 0 As a ﬁnal remark in this section, let us note that, for example, Shishikura states Lemma 6.2 and Theorem 6.4 [Sh] for an open set of maps in a Banach manifold, and the proofs he gives actually imply that Douady’s coordinates vary locally analytically with the map η. We will prove a similar statement further in this paper. 7. Renormalization of maps of the cylinder 7.1. A space of maps of the cylinder and a renormalization transformation In this section we will introduce a new renormalization procedure deﬁned on a space of analytic maps of the cylinder. As it will turn out this renormal- ization is naturally related to the renormalization of critical circle maps, and has some important advantages over the latter. Firstly, let us denote π the natural projection C → C/Z. For an equatorial annulus U ⊂ C/Z let A be the space of bounded analytic maps φ : U → C/Z,such that φ(T) is homotopic to T, equipped with the uniform metric. We shall turn A into a real-symmetric com- −1 plex Banach manifold as follows. Denote U the lift π (U) ⊂ C.The space of ˜ ˜ functions φ : U → C which are analytic, continuous up to the boundary, and ˜ ˜ 1-periodic, φ(z + 1) = φ(z), becomes a Banach space when endowed with the sup ˜ ˇ norm. Denote that space A .For a function φ : U → C/Z denote φ an arbitrarily −1 ˇ ˜ ˇ ˜ chosen lift π(φ(π (z))) = φ.Observe that φ ∈ A if and only if φ = φ − Id ∈ A . U U We use the local homeomorphism between A and A given by U U −1 ˜ ˜ φ → π ◦ (φ + Id) ◦ π to deﬁne the atlas on A . The coordinate change transformations are given by ˜ ˜ φ(z) → φ(z + n) + m for n, m ∈ Z, therefore with this atlas A is a real-symmetric complex Banach manifold. Now let f : U → C/Z be an analytic critical circle map. By deﬁnition, there is a neighborhood of the equator in which 0 is the only critical point of f .Let ˜ ˜ f : U → C be a lift of f . The Argument Principle implies that for >0 small 24 MICHAEL YAMPOLSKY enough, if g˜ ∈ A is real-analytic, with g˜ (0) =−1, g˜ (0) = 0,and || f − g˜|| < , −1 then g = π ◦(g˜+Id)◦π is a critical circle map as well. Let ( f ) be the supremum of such ,and set −1 C =∪ π ◦ (g˜ + Id) ◦ π | g˜ ∈ A , g˜ (0) =−1, g˜ (0) = 0, U f U and ||g˜ − f || < ( f ) . Proposition 7.1. — The space C is a codimension 2 submanifold of A . U U Proof. — Set ˜ ˜ ˜ ˜ B ={φ ∈ A with φ (0) = 0, φ (0) = 0}. U U This is a Banach subspace of A , which has codimension 2 by the Implicit Func- tion Theorem. Select an element ψ ∈ A with ψ (0) =−1, ψ (0) = 0.In the local chart on A deﬁned by −1 ˜ ˜ φ → π ◦ (φ + Id) ◦ π the image of C is the afﬁne subspace ψ +B . U U We shall say that f is a critical cylinder map if f ∈ C for some U ⊂ C/Z. Let us denote by C the real slice of C : C ={ f ∈ C with f (z) = f (z)} ={ f ∈ A , such that f | is a critical circle map}. U T Deﬁnition 7.1. — Given a critical cylinder map f ∈ C let us say that it is cylinder renormalizable, or simply renormalizable, if there exists k>1 and an equatorial annulus V ⊂ C/Z such that following holds: • there exist repelling periodic points p , p of f in U with periods k and a simple arc 1 2 k k l connecting them such that f (l) is a simple arc, and f (l) ∩ l ={ p , p }; 1 2 k k • the iterate f is deﬁned and univalent in the domain C bounded by l and f (l),the −k corresponding inverse branch f | univalently extends to C ; and the quotient of f (C ) f C ∪ f (C )\{ p , p } by the action of f is a Riemann surface conformally isomorphic f f 1 2 to the cylinder C/Z (we will call a domain C with these properties a fundamental crescent of f ); j n(z) ¯ ¯ • for a point z ∈ C with { f (z)} ∩ C =∅,set R (z) = f (z) where f j∈N f C n(z) n(z) ∈ N is the smallest value for which f (z) ∈ C . We further require that there exists a point c in the domain of R such that f (c) = 0 for some m<n(c);and ˆ ˆ if we denote f the projection of R to C/Z with c → 0,then f ∈ C . C V We will say that the new critical circle map f is a cylinder renormalization of f with period k. HYPERBOLICITY OF RENORMALIZATION 25 The relation of the cylinder renormalization procedure to critical circle maps is easy to see: Proposition 7.2. — Suppose f ∈ C is a critical circle map with rotation number ρ( f ) ∈ R\Q . Assume that it is cylinder renormalizable with period q . Then the corresponding ˆ ˆ renormalization f is also a critical circle map with rotation number G (ρ( f )).Also, Rf is n+1 analytically conjugate to R f . Proof. — As follows from the deﬁnition of a fundamental crescent, the in- tersection of the domain C with T is an arc of the form [a, f (a)].The claim follows in an obvious fashion from this observation. The ﬁrst useful property of cylinder renormalization is the following: Proposition 7.3. —Let f ∈ C be renormalizable, with a cylinder renormalization f ∈ C corresponding to the fundamental crescent C ,and let W be any equatorial annulus V f compactly contained in V. Then there is an open neighborhood G ⊂ C of f such that every g ∈ G is renormalizable, with a fundamental crescent C ⊂ U which depends continuously on g in the Hausdorff sense. Moreover, there exists a holomorphic motion χ : ∂C → ∂C over G, g f g such that χ ( f (z)) = g(χ (z)). And ﬁnally, the renormalization gˆ is contained in C . g g W Proof. — Firstly, there is a neighborhood of f in which there exist continuous g g f branches of g -periodic points p , p with p = p ,suchthatbothpointsare re- 1 2 i pelling. Shrinking the neighborhood, if necessary, we may select an analytic family g g of linearizing coordinates w : D → D which conjugate the iterate g to the linear i i g g g g g map z → λ z,with λ = (g ) ( p ) (here D is a topological disk around p ,and i i i i i g g g g w ( p ) = 0). We may further assume that w (D ) = D. Suppose C is bounded by i i i i the curves l , f (l ) terminating at the points p ,and let usdenote z the points f f i k k in ∂D ∩ f (l ) which are obtained last, when moving along f (l ) starting from f f −k the equator, and ζ their preimages by the branches of f deﬁned in D and f f f f ﬁxing p . Note that removing a smooth simple curve γ ⊂ D connecting p to i i i i f f the boundary of D from D we may select a branch of i i f g W = log w i i log λ conjugating f to z → z + 1 in the complementary domain. Let us embed γ g g into an analytic family of curves γ with the same properties, and deﬁne W in i i a similar fashion, selecting the same branch of log for all maps in a neighborhood of f . Now we are prepared to deﬁne C .Let l be the segment of l terminating g f at the points ζ , ζ ,and l the complementary segment containing p .Now let 1 2 i i 26 MICHAEL YAMPOLSKY f g L be the curve l in W -coordinates, and L the parallel translate of this curve i i i i g g terminating at W (ζ ).Weset l to be the union of three curves: l,and l = i g i i g g −1 k (W ) (L ),and deﬁne l = g (l ). By construction, shrinking the neighborhood i i g around f again, if necessary, we may ensure that the curves l , l are disjoint away from the ends. Denote C the domain bounded by l , l , chosen so that g g g → C is Hausdorff continuous. By the Grotzsch inequality, C is a fundamental g g crescent for g. The existense of the desired holomorphic motion is obvious from the construction. It remains to show that G may be chosen so that g ∈ C . It clearly follows −k from Deﬁnition 7.1 that there is a domain D ⊂ C ∪ f (C ) and an iterate n> k f f such that W π (D) (where π : C → C/Z denotes the appropriate projection) f f f n n and f ∈ C is the projection of the iterate f | . Now the iterate g | projects to W D D an analytic map g which is in C for all g sufﬁciently close to f . The next proposition shows that g → g is well-deﬁned: Proposition 7.4. — In the notations of of Proposition 7.3, let C be a different family of fundamental crescents, which also depends continuously on g ∈ G in the Hausdorff sense. Then there exists an open neighborhood G ⊂ G of f , such that the cylinder renormalization of g ∈ G corresponding to C is also gˆ. Proof. — The argument is standard in the study of Douady coordinates; we outline it, and leave the details to the reader. As we have seen above, in a slit neighborhood of p ,the iterate g is conjugate to z → z + 1,and both C and C i g become fundamental half-inﬁnite strips for the orbits of the unit translation. Let us deﬁne a projection π : C → C as follows: in a small neighborhood of the re- pelling point p the point z ∈ C will project to a point z ∈ C , which is its integer i g translate in the log-linearizing coordinate. Outside the linearizing neighborhoods of k k −1 the repelling points, every z ∈ C belongs to a union C ∪ f (C ) ∪ ( f | ) (C ) g g g C g (provided G is chosen small enough). We then set π(z) to be one of the points k k −1 z, f (z), ( f | ) (z) which belongs to C . In the quotient cylinders, π becomes g g aconformal map C/Z → C/Z which conjugates the corresponding cylinder renor- malizations of g. However, it also ﬁxes 0 ∈ C/Z and hence is the identity. We invite the reader to verify, along the same lines, that if t → C is a ho- motopy of fundamental crescents for f , then the corresponding cylinder renormal- izations all coincide. However, a critical cylinder map may be renormalizable in several different ways. Further in this section we will make a canonical choice of cylinder renormalization for a particular class of critical cylinder maps. A key property of the cylinder renormalization is the following: HYPERBOLICITY OF RENORMALIZATION 27 Proposition 7.5. — In the notations of Proposition 7.3, the dependence g → g is an analytic map G → C . Proof. — By the Theorem of Bers and Royden [BR], the holomorphic motion k k χ extends to a holomorphic motion χ : C ∪ f (C ) → C ∪ g (C ) over a smaller g g f f g g open neighborhood G with the same equivariance property. As shown in [MSS], the holomorphic motion induces an analytic family of quasiconformal maps k k k k Ψ : C/Z = (C ∪ f (C ))/f → C/Z = (C ∪ g (C ))/g . g f f g g Applying the theorem of Ahlfors and Bers, we see that the projection π : C ∪ g g k −k g (C ) → C/Z depends analytically on g.Let D ⊂ C ∪ f (C ) and n ∈ N be such g f f n n that W π (D),and f ∈ C is the projection of the iterate f | .The iterate g | f W D D projects to π ◦ g = gˆ ∈ C for all g sufﬁciently close to f , and the claim follows. g W Let us summarize. The cylinder renormalization acts on critical circle maps themselves, rather than on commuting pairs. The reason for introducing commut- ing pairs in the ﬁrst place, was the absense of a canonical afﬁne structure on the smooth circle obtained by glueing the ends of a fundamental domain of the iterate n+1 f .However, if f is cylinder renormalizable with period q , the afﬁne structure on the quotient circle comes from the quotient C /f C/Z.The key here is a fundamental fact of complex analysis, known under a slightly different guise as Liouville’s Theorem: an analytic manifold structure on C/Z is necessarily afﬁne. The second advantage of R over R is closely related to the ﬁrst one: as we cyl have seen, it extends to analytic maps which are not symmetric with respect to the circle, thereby becoming an analytic operator on a complex Banach manifold. In the following section we will show that Epstein critical circle maps are always cylinder renormalizable. 7.2. Cylinders of commuting pairs in the Epstein class In this section we will relate the cylinder renormalization deﬁned above with the renormalization of critical commuting pairs in the Epstein class. The crucial observation is the following: Lemma 7.6. —Let ζ = (η, ξ) be a commuting pair in the Epstein class with χ(ζ) =∞.Denote D the domain of the branched triple covering map η : D → C . η η η(I ) Then the following holds: + − ± • there exist two compex conjugate ﬁxed points p , p of the map η with p ∈±H, η η η and an R-symmetric simple arc l connecting these points such that l ⊂ D and + − η(l) ∩ l ={ p , p }; η η 28 MICHAEL YAMPOLSKY • denoting C the R-symmetric domain bounded by l and η(l) we have C ⊂ D , ζ ζ η + − and the quotient of C ∪ η(C ) \{ p , p } by the action of ζ is a Riemann surface ζ ζ η η homeomorphic to the cylinder C/Z (that is, C is a fundamental crescent of η); • the dependence ζ → C is continuous with respect to the Caratheod ´ ory topology on the Epstein class and the Hausdorff topology on the image. Proof. — Let us ﬁx s>0 such that ζ ∈ E . In view of Lemma 3.2, there exists N = N(s) such that every pair ζ ∈ E with χ(ζ) >N satisﬁes the con- ditions of the existence of Douady coordinates (Lemma 6.2). We may choose as C a fundamental domain for η| , by Theorem 6.4 the choice can be made ζ U continuously. −1 Letus addressthe case χ(ζ) ≤ N(s).Let η : H → H denote the inverse −1 branch which preserves the real axis, η (H) H. By the Denjoy-Wolf Theorem this branch has a unique ﬁxed point p ∈ H ∪ η(I ) which is globally attracting + −1 in H. In this case necessarily Im p >0.Extend η to −H by symmetry. This − + branch has a complex-conjugate ﬁxed point p = p . In view of their unique- η η ness, these points depend continuously on the map η. Note that for a converg- ing sequence of commuting pairs with χ(η) →∞ these ﬁxed points converge to a parabolic ﬁxed point on the real axis. Suppose that the commuting pair ζ does not possess a fundamental cres- cent C . As noted in the proof of Proposition 7.3, in a neighborhood of p the map η becomes a translation in log-linearizing coordinate. Therefore if l is a curve ± ± connecting the ﬁxed points p and the image η(l) \{ p } is disjoint from l,then η η l can be modiﬁed near the ends in such a way that the quotient of the cres- cent bounded by l, η(l) has the conformal type of a bi-inﬁnite cylinder. Thus, for every choice of an R-symmetric curve l connecting the ﬁxed points p the image η(l) \{ p } intersects with l.To ﬁx the ideas, let ξ(0) >0. Consider the maps η ∈ E given by η = η + α for α >0.Let α be the smallest value for which α α 0 the map η has a ﬁxed point, which is then necessarily parabolic. For all α < α α 0 sufﬁciently close to α the map η has a fundamental crescent by Lemma 6.2. 0 α Therefore there exists a value α between 0 and α such that η = η has the ∗ 0 ∗ α following property. For any simple R-symmetric arc l connecting the ﬁxed points −1 of η the intersection (η (l) \{ p }) ∩ l is non-empty, and there is a curve l which ∗ ζ −1 touches η (l) without crossing it. −1 ± ˆ ˆ ˆ Consider such a curve l and denote γ = l ∩ η (l) \{ p }. Again appealing ∗ η to the local picture of the dynamics near the ﬁxed points and near the real axis we seethat wemay choose l for which the set γ is positive distance away from p and R.Let us now ﬁx l for which γ has the smallest Euclidean diameter. Enclose γ into a Euclidean disk E with a diameter equal to diam γ,and take apoint x ∈ γ ∩ ∂E.Let x , x , ... denote the consecutive η -preimages of x lying −1 −2 ∗ HYPERBOLICITY OF RENORMALIZATION 29 in γ . Note that this sequence has to be ﬁnite, since otherwise the set ∪x ⊂ γ is −i −1 compact and invariant under η and thus contains a ﬁxed point, which contradicts ± −1 the fact that p are the only ﬁxed points of η .Let x be the last term in this −n η ∗ −1 sequence. Since η (x )/ ∈ γ , we may slightly modify the curve l near x ,so −n −n that the new curve and its preimage become disjoint in a neighborhood of x . −n ˜ ˆ Proceeding in this fashion, we obtain a curve l which is a small perturbation of l −1 ˜ ˜ such that l ∩ η (l) ⊂ E \ D (x) for some >0. Repeating the procedure a ﬁnite number of times we arrive at a curve l with −1 ± ˇ ˇ diam l ∩ η (l) \ p < diam γ, ∗ η contradictory to the way in which l was selected. Tracing back the assumptions, we see that for any ζ ∈ E with χ(ζ) =∞ the arc l may be chosen with η(l) ∩ l ={ p }. The continuity of the construction is ensured in the same way as in Proposition 7.3. Given ζ = (η, ξ) ∈ E and C as above, denote R (z) the ﬁrst return map ζ C of the crescent C .Let c ∈ T be the only critical point of R and denote ζ ζ C π : C ∪ η(C ) → C/Z the projection speciﬁed by c → 0.Inanobvious fashion C ζ ζ ζ we have: Proposition 7.7. — The projection of the ﬁrst return map −1 f = π ◦ R ◦ π C C C ζ ζ ζ is deﬁned and analytic on an open neighborhood of the circle T,and f | is an analytic C T critical circle map. Let us show that this critical circle map does not, in fact, depend on the particular choice of the fundamental crescent C : Proposition 7.8. —Let C ⊂ D be a different domain satisfying the conditions of ζ ζ Lemma 7.6. Then f | ≡ f | . C T C T Proof. — To ﬁx our ideas, assume that ξ(0) >0.Denote O ⊂ C the smallest full set in the plane containing C ∪ C .The set O is bounded by two curves l , ζ 1 −1 l meeting at p ,let l ∩ R <l ∩ R. By construction, the curves l and η (l ) 2 1 2 1 1 bound a fundamental crescent C . The Denjoy-Wolff Theorem together with the dynamics of η| implies that for every point z ∈ C there is the smallest n(z) ≥ 0 −n(z) −n(z) such that η (z) ∈ C .Deﬁne h(z) ≡ η (z).The value of n(z) is constant on ζ 30 MICHAEL YAMPOLSKY n(z) an open neighborhood of z except for z ∈ η (∂C ) for which the values on the −1 two sides of ∂C differ by 1. Therefore the projection ψ = π ◦ h ◦ π is an ζ C −1 analytic endomorphism of C/Z.Since η is univalent, ψ has no critical points, and since η is monotone, the degree of ψ| is one. Hence, ψ is a conformal R T map. Also, ψ(0) = 0 by construction of the projections π and π . Therefore, C C −1 ψ ≡ Id. By deﬁnition of the ﬁrst return map, ψ ◦ f ◦ψ = f on a neighborhood C C of T,so f = f in that neighborhood. The equality f = f is shown in the C C C C ζ ζ ζ same way, and the claim follows. In view of the last lemma, on a neighborhood of T we obtain a well-deﬁned analytic critical circle map f ≡ f , ζ C which we will refer to as the cylinder map of ζ . In Figure 4 we illustrate the action of the map f near T. Schematically shown is the preimage of the circle T ⊂ C/Z which consist of the circle itself, together with analytic arcs emanating from 0 at angles kπ/3 to T. Analytic continuations of these preimages may eventually branch further, as shown. FIG. 4. 7.3. The cylinder renormalization operator Proposition 7.9. —For every s ∈ (0, 1) there exists an equatorial neighborhood U = U(s) such that for every commuting pair ζ ∈ E with χ(ζ) =∞ the cylinder map f is s ζ in C . U HYPERBOLICITY OF RENORMALIZATION 31 Proof. — First, denote mod ζ = sup mod U over all equatorial annuli U such that f ∈ C . To prove the claim, let us assume the contrary. Then there is ζ U a sequence ζ ∈ E with χ(ζ ) =∞ such that mod(ζ ) → 0. Using compactness n s n n of E as stated in Lemma 3.2 we may pass to a subsequence ζ → ζ ∈ E .The s n s height of the commuting pair ζ is necessarily inﬁnite. The dynamics of ζ produces an analytic return map h from an equatorial neighborhood in the repelling Fatou cylinder of η to the attracting Fatou cylinder. Again by Lemma 3.2, there exists δ = δ(s) >0 such that h ∈ C with modU> δ. By Theorem 6.4 for any ζ ζ U n sufﬁciently close to ζ , mod ζ > δ, and we have arrived to a contradiction. For the remainder of this paper let us ﬁx the value of s to be the universal constant from Lemma 3.4. Let us say that two pairs ζ , ζ in E are conformally 1 2 s equivalent, ζ ∼ ζ ,if χ(ζ ) = χ(ζ ) =∞ and f | ≡ f | . The name of the equiv- 1 conf 2 1 2 ζ T ζ T 1 2 alence relation is explained by the following: Proposition 7.10. — Suppose ζ ∼ ζ . Then there exists a conformal map ψ from 1 conf 2 a neighborhood of I to a neighborhood of I which conjugates the two Epstein commuting ζ ζ 1 2 pairs. Conversely, assume that there is a conjugacy ψ, whose domain includes a fundamental crescent C .Then ζ ∼ ζ . ζ 1 conf 2 Proof. — The ﬁrst claim is quite obvious, since the identity map of C/Z lifts via the projections π , π to a conformal conjugacy between the two commut- C C ζ ζ 1 2 ing pairs. On the other hand, any such conjugacy ψ pojects to a conformal map h between equatorial annuli, which conjugates f , f .Ifthe domain of ψ contains ζ ζ 1 2 C ,then h extends to a conformal endomorphism of C/Z.Since h(0) = 0,in this case h ≡ Id. Recall that S ={ζ ∈ C| χ(ζ) = r}. The equivalence relation ∼ is clearly r conf closed. By Lemma 7.6 we have: Proposition 7.11. —For every r ∈ N the correspondence ι : (S ∩ E )/ ∼ → C r s conf given by [ζ ] → f is continuous. ∼ ζ conf Moreover, Proposition 7.12. —For every s ∈ (0, 1) and r ∈ N there exists c = c(s, r) >0 such 0 R that the following holds. If we equip the Epstein class E with C -distance, and C with the sup-distance, then for every ζ ,ζ ∈ E ∩S we have dist 0 (ζ ,ζ ) ≥ c(s, r) dist 0 ( f , f ). 1 2 s r C 1 2 C ζ ζ 1 2 Proof. — For ζ = (η, ξ) ∈ E let φ denote the uniformizing coordinate C → C/Z,and Φ : C → C be its lift. By Lemma 7.6 the crescent C moves ζ ζ ζ continuously in E . On the other hand, by Lemma 3.2, E is Caratheodory com- −1 pact, and hence the domain C ∪ η(C ) ∪ η (C ) has a bounded shape near ζ ζ ζ 32 MICHAEL YAMPOLSKY −1 C ∩ R.Since Φ conformally extends to C ∪ η(C ) ∪ η (C ), we may apply the ζ ζ ζ ζ Koebe Distortion Theorem to it to conclude that the distortion of Φ on C ∩ R is bounded by a constant which depends only on s and r.Moreover, by real apriori bounds the derivative Φ is bounded by const(s, r).Noting that f | is the ζ T projection of the composition η ◦ ξ , we have the desired estimate. Proposition 7.13. —Let U be as in Proposition 7.9. There exists N = N(U) ∈ N such that the following holds: (I) suppose ζ ∈ E is at least N-times renormalizable. Then the renormalization R ζ ∈ E ; (II) also R ζ has a holomorphic pair extension H with universal bounds as in Theo- rem 5.4; (III) denote H the holomorphic pair extension of pR ζ which is the linear rescaling −l of H . Then there exists an iterate η which conformally conjugates the pair H to a holomorphic pair G with ∆ C for some fundamental crescent C ; G ζ ζ (IV) moreover, π : ∆ → C/Z ∩ U; C G (V) ﬁnally, there is a fundamental crescent C N−1 compactly contained in U (cf. pR f Figure 4). Proof. — By real apriori bounds, eventually all renormalizations of a com- muting pair ζ ∈ E lie in E . The existense of N as in (I) follows from Lemma 3.2. Statement (II) is a consequence of Theorem 5.4 and Lemma 3.2. Claims (III) and (IV) follow since the range of the holomorphic pair H is commensurable with I by Theorem 5.4. Finally, (V) follows from Lemma 7.6 and Lemma 3.2. Deﬁnition 7.2. — For the remainder of the paper ﬁx U as in Proposition 7.9 and set M = C , N = N(U). Condition (V) of Proposition 7.13 implies that f is a renormalizable critical cylinder map. Let f be its renormalization corresponding to the crescent C N−1 .We pR f will call it the cylinder renormalization of f , and write f ≡ R f . cyl ζ It follows from the ﬁrst condition of Proposition 7.13 that R f ∈ C . Therefore, by cyl ζ U Lemma 7.3, for every pair ζ as above, the tranformation f → R f extends to an open ζ cyl ζ neighborhood Y ⊂ C as an analytic operator Y → C . We shall call this operator the U U cylinder renormalization operator. The deﬁnition of R implies the following connection with the usual renor- cyl malization operator: Proposition 7.14. — In the above notation, we have R f ≡ f N . cyl ζ R ζ HYPERBOLICITY OF RENORMALIZATION 33 tq Proof. — By condition (V) of Proposition 7.13, there is an iterate f ,for −1 some t ∈ Z, which conformally maps π (C N−1 ) to C N .Thisconformal map R f R ζ C ζ projects to a conjugacy ψ : C/Z → C/Z between R f and f N .Since ψ is cyl ζ R ζ conformal, and ψ(0) = 0, necessarily ψ ≡ Id. Moreover, Proposition 7.15. —If ζ , ζ are at least N-times renormalizable elements of E ,and 1 2 s N N ζ ∼ ζ ,then R ζ ∼ R ζ . Hence the action of R is well-deﬁned on the quotient space 1 conf 2 1 conf 2 E / ∼,and s conf ι ◦ R = R ◦ ι. cyl Proof. — The claim follows from condition (V) of Proposition 7.13, similarly to the preceding Proposition. 8. Local stable manifold of a periodic point of R cyl Let a commuting pair ζ be a periodic point of the renormalization opera- tor R. As follows from Lemma 3.4, ζ ∈ E . Naturally, ζ is also a periodic point N Nm of R , and we may ﬁx the smallest m ∈ N such that R (ζ) = ζ . As seen from Proposition 7.15, the critical circle map f = f ∈ M is a periodic point of the cylinder renormalization operator: ˆ ˆ R f = f . cyl ˆ ˆ Letusﬁxa periodic point f .Set ρ = ρ( f ),and deﬁne D ={ f ∈ M,such that ρ( f ) = ρ}. Again from Proposition 7.15 we have Proposition 8.1. — There exists a neighborhood Y of f in M such that for every im im f ∈ Y ∩ D , R f is deﬁned for all i,and R f → f uniformly in Y. cyl cyl Proof. — Assume that the statement of the proposition does not hold. Since R is deﬁned in an open neighborhood of f , the assumption implies that there cyl exists δ >0 such that for every i ∈ N and every >0 there exists f such that i, im ˆ ˆ dist( f , f ) < ,and dist( f , R f ) > δ. Select a sequence → 0 and denote i, i, i cyl f = f .Let λ be the length of the domain of (imN − 1)-st pre-renormalization i i, i imN−1 of f : λ =|I |, and consider the inﬁnite sequence i i pR f −1 imN−1+j T = λ pR f (λ z) . i i i j=−imN−1 By complex apriori bounds (Theorem 5.4) for each i there exists such N that for all n ≥ N the renormalization R f extends to a holomorphic commuting pair in i 34 MICHAEL YAMPOLSKY H(µ) with a universal bound µ.By Remark 5.1,the valueof N can be chosen uniformly for all i. By Lemma 5.3 we may select a subsequence {i } such that the sequences T converge to a limiting renormalization tower T with complex apriori bounds. Denote T the limiting renormalization tower corresponding to the periodic point ζ . By Proposition 7.12, the distance between the base pairs of T and T is at least const ·δ. This contradicts the Tower Rigidity Theorem. Below we shall demonstrate that the local stable set of f is a submanifold of M: Theorem 8.2. — There is an open neighborhood W ⊂ M of f such that D ∩ W is a smooth submanifold of M of codimension 1. Fix an equatorial neighborhood U U such that f analytically extends to U . As before denote p /q the reduced form of the k-th continued fraction ˆ k k convergent of ρ.Furthermore,deﬁne D as the set of g ∈ M for which ρ(g) = p /q k k k and 0 is a periodic point with period q . As follows from the Implicit Function Theorem, this is a codimension 1 submanifold of M. As seen in the previous section, the tangent space of a point in C may be identiﬁed with B ,and that U U R R of a point in M with the real slice B . Let us deﬁne a cone C ∈ B as U U C = v ∈ B , such that inf v(x) >0 . x∈R Lemma 8.3. —Let f ∈ D , and denote T D ⊂ B the tangent space at this point. k f k Then T D ∩ C =∅. f k Proof. — Let v ∈ C and suppose { f }⊂ M is a one-parameter family such that −1 −1 π ◦ f ◦ π = π ◦ f ◦ π + tv + o(t). −1 −1 k q Then for sufﬁciently small values of t, π ◦ f ◦π > π ◦ f ◦π.Hence f (0) = f (0) and thus f ∈ / D . t k Now let f be as above. Elementary considerations of the Intermediate Value Theorem imply that for every k there exists a value of θ ∈ (0, 1) such that the map f = R ◦ f ∈ D .Moreover, if we denote θ the angle with the smallest θ θ k k absolute value satisfying this property, then θ → 0.Hence, for all k large enough R (U ) ⊃ U,so f ∈ M.Set f = f and let T = T D ⊂ B .Fix v ∈ C.By θ ˆ θ k θ k f k k f k k k U Lemma 8.3 and the Hahn-Banach Theorem there exists >0 such that for every R ∗ k there exists a linear functional h ∈ (B ) with ||h || = 1,such that Ker h = T k k k k U HYPERBOLICITY OF RENORMALIZATION 35 and h (v) > . By the Alaoglu Theorem, we may select a subsequence h weakly k n R ∗ converging to h ∈ (B ) . Necessarily v ∈ / Ker h,so h ≡ 0.Set T = Ker h. Proof of Theorem 8.2. — By the above, we may select a splitting B = T⊕v·R. R R Denote p : B → T the corresponding projection, and let ψ : M → B be U U a local chart at f . Lemma 8.3 together with the Mean Value Theorem imply that p ◦ ψ : D → T is an isomorphism onto the image, and there exists an open neighborhood U of the origin in T,suchthat p ◦ ψ(D ) ⊃ U.Since thegraphs D are analytic, we may select a C -converging subsequence D whose limit is k k a smooth graph G over U . Necessarily, for every g ∈ G, ρ(g) = ρ.Aswehave seen above, every point g ∈ D in a sufﬁciently small neighborhood of f is in G, and thus G is an open neighborhood in D . 9. Proof of the main result As in the previous section, let f : U → C/Z,where U U,beaperiodic ˆ ˆ f f point of R with period m.Set ρ = ρ( f ) and let W be as in Theorem 8.2. cyl Denote T ≡ T (D ∩ W) the tangent space to the codimension one submanifold ˆ ρ ˆ ˆ at f ,and let L be the differential of R at f : cyl m R R L = D R : B → B , L : T → T f cyl U U Recall that a continuous linear operator on a Banach space is called compact if it maps the closed unit ball of the space onto a compact set. This condition is equivalent to the image of every closed bounded set being compact. m R R Proposition 9.1. — The operator L = D R : B → B is compact. f cyl U U Proof. — Denote B the unit ball in B and let v ∈ B . By deﬁnition of 1 1 R , the image L v is an analytic vector ﬁeld on U U.Let C U be cyl ˆ ˆ f f the fundamental crescent corresponding to R f ,and φ = π : C → C/Z the cyl ˆ ˆ f f −1 uniformizing coordinate. The ﬁrst return map R of φ (U) ⊂ C consists of a ﬁnite collection of iterates f . By compactness considerations, the vector ﬁeld v which is the perturbation of each of these iterates induced by v has norm bounded by −1 a constant C independent of v.Since φ is also bounded in φ (U), there exists D>0 independent of v such that ||L v| || <D. Since the analytic functions in U whose norms in U are bounded by D form a normal family in U, the image L (B ) is compact. Hyperbolicity: stable direction. — By Proposition 8.1, sup ||L | || < ∞. k 36 MICHAEL YAMPOLSKY Let us show that for some iterate the norm is strictly less than one: Proposition 9.2. — There exists k ∈ N such that ||L | || <1. Proof. — Assume the contrary. Then the operator L | has a unit spec- tral radius. The discreteness of the spectrum of a compact operator implies that Sp(L | ) ∩ B =∅, and moreover, since every non-zero element of the spectrum T 1 of a compact operator is an eigenvalue, there is v ∈ T such that L (v) = v.The s c s space T may then be split into a direct sum E ⊕ E such that E , the stable space, is a closed L -invariant subspace such that ||L | s || <1 for some ∈ N, and E , the central space, is also L -invariant, ﬁnite dimensional, and for every c c k v ∈ E , inf ||L v|| >0.The space E is spanned by solutions of (T − λ Id) v = 0, ∈N k ∈ N, λ ∈ B . If we choose a small perturbation f ∈ D ∩ W of f in the direction 1 ρ c m of E close enough to f , then the dynamics of R on f is going to be dominated cyl km by the central part L | . In particular, the rate of convergence R f −→ f can cyl k→∞ not be geometric, contrary to Theorem 4.3. Hyperbolicity: unstable direction. — As before, let B , such that inf v(x) >0 . C = v ∈ x∈R The key properties of C are listed in the next lemma: Lemma 9.3. (I) The cone C is renormalization-invariant: L : C → C , (II) Moreover, there exists α >0 and k ∈ N such that for any vector ﬁeld v ∈ C inf L (v(x)) > (1 + α) inf v(x), x∈R x∈R (III) Finally, if v ∈ B belongs to the closure of C,and v = 0, then there exists ∈ N for which L (v) ∈ C . −1 ˜ ˆ Proof. — Let f = π ◦ f ◦ π . As follows from an elementary computation, if ˜ ˜ f (x) = f (x) + tv(x) + o(t),then 2 2 ˜ ˜ f (x) = f (x) + tv (x) + o(t) (9.1) ˜ ˜ ˜ ˜ = f (x) + t( f ( f (x))v(x) + v( f (x))) + o(t), and more generally, if we write n n ˜ ˜ f (x) = f (x) + tv (x) + o(t), (9.2) n−1 n−1 then v (x) = f ( f (x))v (x) + v( f (x)). n n−1 HYPERBOLICITY OF RENORMALIZATION 37 Let us denote C U the fundamental crescent of f corresponding to the k-th cylinder renormalization R f ,and J = C ∩ T.Let π (x) be the corresponding k k k cyl uniformizing coordinate C → C/Z,and Φ : C → C its lift. In the local chart k k n −1 given by π ,the k-th cylinder renormalization R f is represented by the iterate k cyl mk g = f for some m ∈ N. To prove the ﬁrst claim, observe that by (9.2), (9.3) inf v (x) ≥ inf v(x) >0. x∈R x∈R The image −1 (9.4) L v =[(Φ ) ◦ g · (v | )]◦ (Φ ) . k k q J k mk k Since (Φ ) >0, (I) follows. To prove(II), notethat by the real apriori bounds, the sizes of the intervals J decrease geometrically with k. On the other hand, applying the Koebe Distortion Theorem to the conformal extension of Φ to C and its two neighboring iterates, k k we see that the distortion of Φ on the interval g ( J ) is bounded uniformly in k. k k k Hence, there exists α >0 and l ∈ N such that for all k ≥ l (Φ ) | >1 + α, k g ( J ) k k and (II) follows from (9.4). Finally, to see (III), observe that if v(x) ≡ 0, then there exist , n ≤ q such that v(x) >0 for every x ∈ f ( J ). The claim now follows from (9.2). We conclude: Proposition 9.4. — There exists an eigenvector v ∈ C : L (v ˆ) = δv ˆ, where |δ| >1. Proof. — Part (II) of the previousLemma impliesthatthe spectral radius R (L ) >1. Since the spectrum of a compact operator is discrete, and its every Sp non-zero element is an eigenvalue, the claim follows. Universality: Proof of Corollary 4.5. — The second claim is an obvious corollary of the main theorem, the proof of the ﬁrst one follows. Denote ζ ∈ E the ﬁxed point of ∗ s R with ρ(ζ ) = ( 5 − 1)/2,and set f = ι(ζ ), deﬁned in some annulus U ⊃ U.By ∗ ∗ ∗ f Theorem 4.4, there exists m ∈ N such that f is a hyperbolic ﬁxed point of R with ∗ cyl loc one-dimensional unstable direction. Let W be its local stable manifold, set M = mN, and denote δ >1 the modulus of the unstable eigenvalue. Set P ={ f ∈ C , such that ρ( f ) = [1, ..., 1], and f has a periodic orbit with eigenvalue ± 1}. 38 MICHAEL YAMPOLSKY By the Implicit Function Theorem, P is a local submanifold of C of codi- mension one. Moreover, in a neighborhood of f , R : P → P .Let C be ∗ cyl n n−M a subcone of C with C ∩ B compact. Then by Theorem 4.4, there exists an open neighborhood W ⊂ C of f and K ∈ N,such that for k> K, P ∩ W is U ∗ loc agraph over W transversal to C at every point. Moreover, for every n> K the ∞ loc M sequence {P } converges to W at a geometric rate δ . For our purposes n+lM l=1 s we will set C = D R (C ). ∗ cyl Let V ⊃ T be an equatorial annulus chosen so that f ∈ C .By compex µ V apriori bounds, there exists N ∈ N such that for all n> N the renormaliza- 1 1 tion R f extends to a holomorphic commuting pair with universal bounds. Since R f → ζ ∈ E and by Lemma 7.6, there exists N ≥ N such that for all k> N µ ∗ s 2 1 2 the map f ∈ C is cylinder-renormalizable with period q , and the corresponding µ V k cylinder renormalization g is in C . Fix some such k. By Proposition 7.3, there k U is a neighborhood G of f in C such that the cylinder renormalization f → g µ V µ k ∗ ∗ extends to an analytic operator Z : G → C .Provided G is sufﬁciently small, there t t is an iterate R (G) ⊂ W.Denote {h }⊂ C the smooth family R (Z({ f }∩ G)). µ U µ cyl cyl R R Let u ∈ B be the tangent vector to the family f at f ,and v ∈ B its image µ µ V ∗ U D (R ◦ Z)u. As in Lemma 9.3 we have inf D Zu > 0 and hence v ∈ C . f f µ cyl µ ∗ ∗ Note that the curve {h for µ ∈ I } for i large is bounded by a pair of points µ i+lM ± ± f ∈ P ,where n = i + lM − k − tM, and the claim follows from the transversality n n property of P . 10. Concluding remarks Let us conclude by discussing some remaining open problems in the ﬁeld. Of primary importance to us is bringing Lanford’s Program to a completion. Passing from the hyperbolicity of periodic orbits of R to the hyperbolicity of the whole cyl renormalization horseshoe requires, as a preliminary step, endowing the local sta- ble sets of the points in the horseshoe with the structure of analytic submani- folds. In addition, the case of unbounded type necessitates analyzing the situation when the periods of renormalization grow without a bound, and showing that the rate of convergence to the horseshoe remains uniformly bounded in this situation. We recently carried out the relevant analysis in a joint paper with A. Epstein [EY]. We extend the result of this paper to the whole horseshoe in a forthcoming work [Ya3]. Commonly, renormalization convergence results in one-dimensional dynamics are connected with rigidity properties of the renormalized maps. The problem 1+α relevant to our study is a C rigidity question: When is the conjugacy between two critical circle maps with the same irrational rotation 1+α number, which maps 0 to 0, C -smooth? HYPERBOLICITY OF RENORMALIZATION 39 The study of this question is the main subject of the papers of de Faria and de Melo [dFdM1,dFdM2]. These authors demonstrate that the conjugacy has the desired smoothness in the case when both maps are analytic, and the type of 1+α the rotation number is bounded. They prove that, with these restrictions, the C rigidity is equivalent to the geometric rate of convergence of renormalizations, and derive their result from their Theorem 4.3. On the other hand, they show that if both of the conditions are relaxed, the statement does not hold. As we described above, we expect our methods to bring Lanford’s Program to a completion, and in particular, to extend Theorem 4.3 to unbounded types. It is then an intriguing problem, whether the answer to the above question is positive when the two maps are analytic, but the rotation numbers are unbounded. The situation when the type is bounded, but the maps are only smooth also requires study. In this paper we have introduced a new renormalization for critical circle maps. We hope that the reader has been convinced that apart from enabling us to solve the hyperbolicity problem, this new construction is more natural. It would be instructive therefore to re-do the proof of Theorem 4.1 using R instead of R. cyl This mainly requires understanding the geometry of a periodic point of R to cyl deﬁne an appropriate analogue of the Epstein class, and ﬁnding the corresponding replacements of de Faria’s holomorphic commuting pairs. For the moment, the only situation when we are able to carry out the whole theory without recourse to commuting pairs is the degenerate case of parabolic renormalization, studied in [EY]. Let us also comment that our method should be applicable to the study of universality of Siegel disks (see e.g. [MP]) in the situation when complex apriori bounds exist (such as Siegel quadratics of bounded type, cf. [Ya1]). This problem is analogous to the study of renormalization of non-real quadratic polynomials and the related aspects of the geometry of the Mandelbrot set, and should lead to the construction of hyperbolic non-real periodic orbits of R . cyl REFERENCES [BR] L. 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Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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Abstract

HYPERBOLICITY OF RENORMALIZATION OF CRITICAL CIRCLE MAPS by MICHAEL YAMPOLSKY The renormalization theory of critical circle maps was developed in the late 1970’s–early 1980’s to explain the occurence of certain universality phenomena. These phenomena can be observed empirically in smooth families of circle home- omorphisms with one critical point, the so-called critical circle maps, and are analo- gous to Feigenbaum universality in the dynamics of unimodal maps. In the works of Ostlund et al. [ORSS] and Feigenbaum et al. [FKS] they were translated into hyperbolicity of a renormalization transformation. The ﬁrst renormalization transformation in one-dimensional dynamics was constructed by Feigenbaum and independently by Coullet and Tresser in the set- ting of unimodal maps. The recent spectacular progress in the unimodal renormal- ization theory began with the seminal work of Sullivan [Sul1,Sul2,MvS]. He intro- duced methods of holomorphic dynamics and Teichmul¨ ler theory into the subject, developed a quadratic-like renormalization theory, and demonstrated that renor- malizations of unimodal maps of bounded combinatorial type converge to a horse- shoe attractor. Subsequently, McMullen [McM2] used a different method to prove a stronger version of this result, establishing, in particular, that renormalizations converge to the attractor at a geometric rate. And ﬁnally, Lyubich [Lyu4,Lyu5] constructed the horseshoe for unbounded combinatorial types, and showed that it is uniformly hyperbolic, with one-dimensional unstable direction, thereby bringing the unimodal theory to a completion. The renormalization theory of circle maps has developed alongside the uni- modal theory. The work of Sullivan was adapted to the subject by de Faria, who constructed in [dF1,dF2] the renormalization horseshoe for critical circle maps of bounded type. Later de Faria and de Melo [dFdM2] used McMullen’s work to show that the convergence to the horseshoe is geometric. The author in [Ya1,Ya2] demonstrated the existence of the horseshoe for unbounded types, and studied the limiting situation arising when the combinatorial type of the renormalization grows without a bound. Despite the similarity in the development of the two renormalization theories up to this point, the question of hyperbolicity of the horseshoe attractor presents a notable difference. Let us recall without going into details the structure of the argument given by Lyubich in the unimodal case. The ﬁrst part of Lyubich’s work was to to endow the This work was partially supported by NSF Grant DMS-9804606. 2MICHAELYAMPOLSKY ambient space of the renormalization transformation with the structure of a complex- analytic manifold, with respect to which renormalization is an analytic operator. He then showed that the stable sets of periodic points of this operator are codimension one submanifolds and used an argument based on McMullen’s Tower Rigidity Theorem and an inﬁnite dimensional version of the Schwarz Lemma to show that renormaliza- tion is a strict contraction in the stable direction. The second part of Lyubich’s work was to show the existence of an unstable direction. This is a fundamentally more dif- ﬁcult part of the proof, based on Lyubich’s Rigidity Theorem [Lyu3] and an elegant argument relating non-hyperbolicity of the renormalization operator to certain topo- logical properties of its orbits. In contrast, in the case of critical circle maps it has been well-known how one may construct an unstable direction for the renormalization. The difﬁculty arises with the ﬁrst part of the program. There is no obvious way to turn the renormalization into an analytic operator. Indeed, the deﬁnition of renormalization for critical circle maps essentially employs real symmetry, and resists complexiﬁ- cation. To circumvent this obstacle, in this paper we introduce a different renor- malization construction, a renormalization of critical cylinder maps. This construction is strongly motivated by a so-called parabolic renormalization, a degenerate case of renormalization, which we studied in detail in [Ya2]. There is a natural functorial relation between the renormalization of cylinder maps and the usual one which allows us to transfer the results back and forth. By virtue of its construction, the cylinder renormalization is an analytic operator on a Banach manifold. We then show that the stable sets of the periodic orbits of this operator are codimension one submanifolds, and prove that every such orbit is hyperbolic. Our result gives a rigorous mathematical explanation of the universality phenomena, settling this well-studied problem. The structure of the paper is as follows. In §1weintroducecritical circle maps. We discuss the universality phenomena and deﬁne renormalization in §2. In §4 we formulate the Renormalization Hyperbolicity Conjecture (Lanford’s Program) and state our results. In the following section we discuss some of the constructions from de Faria’s and our own earlier work. In §6 we introduce parabolic renormal- ization, mainly as a motivation for futher discussion. We then deﬁne the cylinder renormalization operator in §7 and proceed to prove our main theorems. Acknowledgements I had many fruitful discussions with Adam Epstein on the subject of parabolic renormalization of critical circle maps and related topics, and I wish to thank him for generously sharing his mathematical insights with me. I am also grateful to Folkert Tangerman for explaining to me the construction of the unstable direc- HYPERBOLICITY OF RENORMALIZATION 3 tion some years ago. I would like to express my gratitude to Mikhail Lyubich for the many conversations we had as this paper developed, for his advice, patience, and constant encouragement. I want to thank Welington de Melo for the helpful criticism of an earlier version of this work. The bulk of this work was conducted at the Yale Mathematics Department, and I wish to thank my colleagues there for making my postdoctoral experience so enjoyable. My special thanks go to Pe- ter Jones for his guidance. Thanks are also due to IHES where this paper was completed. Finally, I wish to thank the referee for the helpful comments. 1. Preliminaries Some notations. — We use dist and diam to denote the Euclidean distance and diameter in C. We shall say that two real numbers A and B are K-commensurable −1 for K>1 if K |A|≤ |B|≤ K|A|. The notation D (z) will stand for the Euclidean disk with the center at z ∈ C and radius r. The unit disk D (0) will be denoted D. The plane (C \ R) ∪ J with the parts of the real axis not contained in the interval J ⊂ R removed will be denoted C .By the circle T we understand the afﬁne manifold R/Z, it is naturally identiﬁed with the unit circle S = ∂D.The real translation x → x + θ projects to the rigid rotation by angle θ , R : T → T.For two points a and b in the circle T which are not diametrically opposite, [a, b] will denote the shorter of the two arcs connecting them. As usual, |[a, b]| will denote the length of the arc. For two points a, b ∈ R, [a, b] will denote the closed interval with endpoints a, b without specifying their order. The cylinder in this paper, unless otherwise speciﬁed will mean the afﬁne manifold C/Z. Its equator is the circle {Im z = 0}/Z ⊂ C/Z. A topological annulus A ⊂ C/Z will be called an equatorial annulus, or an equatorial neighborhood, if it has a smooth boundary and contains the equator. By “smooth” in this paper we will mean “of class C ”, unless another de- gree of smoothness is speciﬁed. The notation “C ” will stand for “real-analytic”. Critical circle maps. — A critical circle map is an orientation preserving auto- morphism of T of class C with a single critical point c. A further assumption is made that the critical point is of cubic type. This means that for a lift f : R → R of a critical circle map f with critical points at the integer translates of ¯ c, ¯ ¯ f (x) − f (¯ c) = (x − ¯ c) (const + O(x − ¯ c)). We note that all the renormalization results will hold true if in the above deﬁnition “3” as the order of smoothness and the order of the critical point is replaced by any other odd number. To ﬁx our ideas, we will always place the critical point of f at 0 ∈ T. 4MICHAELYAMPOLSKY Being a homeomorphism of the circle, a critical circle map f has a well- deﬁned rotation number, denoted ρ( f ). Itisuseful torepresent ρ( f ) as a contined fraction with positive terms (1.1) ρ( f ) = r + r + r + ··· Further on we will abbreviate this expression as [r , r , r , ...] for typographical 0 1 2 convenience. Note that the numbers r are determined uniquely if and only if ρ( f ) is irrational. In this case we shall say that ρ( f ) (or f itself ) is of the type bounded by B if sup r ≤ B; it is of a periodic type if the sequence {r } is periodic. Recall i i k k that an iterate f (0) is called a closest return of the critical point if the arc [0, f (0)] contains no iterates f (0) with i<k. By a classical result of Poincar´eevery circle homeomorphism f with an irrational rotation number is semi-conjugate to the rigid rotation R . Yoccoz [Yoc] has shown that in the case when f is a critical ρ( f ) circle map. This implies, in particular, that if we denote {p /q } the sequence of m m best rational approximations of ρ( f ) obtained as the truncated continued fractions p /q =[r , r ,..., r ], then the iterates { f (0)} are closest returns of 0.Set m m 0 1 m−1 I ≡[0, f (0)]. An important one-parameter family of examples of critical circle maps, the so-called standard (or Arnold’s) maps, is obtained as follows. Given any θ ∈ R the map A (x) = x + θ − sin 2π x 2π commutes with the unit translation: A(x + 1) = A(x) + 1. Therefore it has a well- deﬁned projection to an endomorphism of the circle. We denote this endomor- phism f , although in reality it only depends on the class θ mod Z.Anelementary computation shows that every A is strictly monotone, and has critical points at integer values of x, all of cubic type. Therefore, each f is a critical circle mapping. The considerations of monotone dependence on the parameter imply that θ → ρ(θ) ≡ ρ( f ) is a continuous monotone map of T onto itself. Whenever t ∈ T is irrational, −1 −1 ρ (t) is a single point. For t = p/q the set ρ (t) is a closed interval, for every parameter value in this interval the homeomorphism f has a periodic orbit with a combinatorial rotation number p/q. This orbit has eigenvalue one at the two endpoints of the interval. HYPERBOLICITY OF RENORMALIZATION 5 2. Universality and renormalization 2.1. The golden-mean universality phenomenon The raison d’etre of the renormalization theory of critical circle maps is a uni- versality phenomenon numerically observed as early as in mid-1970’s in one-par- ameter families like the Arnold’s family described above (see [FKS], [ORSS]). For µ ∈ T let f be a one-parameter family of critical circle maps whose dependence on the parameter is smooth and strictly monotone, with ρ( f ) = 0. Let us denote by I (x) the set of parameter values µ with ρ( f ) = x.It isnot difﬁcult to show that I (x) is a single point for irrational values of x;inthe case when x is rational, I (x) is a closed interval. Consider now the intervals I ≡ I ([r , r ,..., r ]) with r = r = ··· = r = 1. The maps with parameter values 0 1 n−1 0 1 n−1 in I possess periodic orbits with periods q ,where p /q is the n-th convergent of n n n n the inﬁnite continued fraction [1, 1, 1, 1, ...]. The ﬁrst numerical observation one makesisthatthere is a particularnumber δ ≈ 2. 83361 such that the length of −n the interval I behaves asymptotically for large n as const · δ . The rate of the geometric decay δ is universal, that is independent on the chosen one-parameter family f . The second observation applies to the limit of the intervals I , i.e. the µ n parameter value µ for which the map f possesses the golden mean rotation 1 µ number 5 − 1 ρ( f ) = =[1, 1, 1, 1, ...]. Denoting by λ the scaling ratio of the length of the closest return arcs ◦q ◦q n n−1 λ = f (c), c c, f (c) , µ µ 1 1 we observe that λ converge to a constant λ ≈ 0. 7760513.The number λ is n ∗ ∗ again independent of the family. Let us remark, that if we consider homeomorphisms of the circle with a non- ﬂat critical point of the order other than three, the universal constants will change. Using the terminology of statistical physics, the order of the critical point speciﬁes the universality class. 2.2. Deﬁnition of renormalization of critical circle maps The strong analogy with universality phenomena in statistical physics and with the already discovered Feigenbaum-Collett-Tresser universality in unimodal maps, led the authors of [FKS] and [ORSS] to explain the existence of the uni- versal constants by introducing a renormalization operator acting on critical circle maps. The deﬁnition is by no means straightforward. We shall arrive at it after 6MICHAELYAMPOLSKY a brief discussion, which will be helpful in understanding the motivation of the resultsin thispaper. The general strategy of deﬁning a renormalization of a given dynamical sys- tem F : X → X is as follows (c.f. [Lyu2]). Select a proper subset Y of X.For apoint x ∈ Y whose forward orbit under F returns to Y denote R (x) ∈ Y the ﬁrst such return. Provided that it is true for every point in Y, this deﬁnes the ﬁrst return map R : Y → Y.Let A : Y → A(Y) be an afﬁne map, such that the size of A(Y), appropriately understood, is the same as the size of the original −1 space X. The rescaled ﬁrst return map A ◦ R ◦ A is the renormalization of F. For the role of the set Y in the case of a critical circle map f : T → T one takes the union of arcs A = I ∪ I for some m>1, where as before I m m m+1 m stands for [0, f (0)]. This choice was made in [ORSS] and [FKS], and has been followed since, although a different choice would have led to the same results. The q q m m+1 ﬁrst return map R : A → A is deﬁned piecewise by f on I and by f m m m m+1 on I .Toview R as a critical circle map we may identify the neighborhoods m m q q q −q m m+1 m+1 m of points f (0) and f (0) by the iterate f . This identiﬁcation transforms the arc A into a C -smooth closed one-dimensional manifold A , R projects to m m m ˜ ˜ ˜ a smooth homeomorphism R : A → A with a critical point at 0.However,the m m m manifold A does not possess a canonical afﬁne structure; the choice of a smooth identiﬁcation φ : A → R/Z gives rise to a plethora of different critical circle maps, all smoothly conjugate. As we can see from the above discussion, the space of critical circle maps is ill-suited to deﬁne renormalization. The authors of [ORSS] and [FKS] circum- vented this difﬁculty by replacing critical circle maps with different objects: Deﬁnition 2.1. —A commuting pair ζ = (η, ξ) consists of two C -smooth orien- tation preserving interval homeomorphisms η : I → η(I ), ξ : I → ξ(I ),where η η ξ ξ (I) I =[0,ξ(0)], I =[η(0), 0]; η ξ (II) Both η and ξ have homeomorphic extensions to interval neighborhoods of their re- spective domains with the same degree of smoothness, which commute, η ◦ ξ = ξ ◦ η; (III) ξ ◦ η(0) ∈ I ; (IV) η (x) = 0 = ξ (y),for all x ∈ I \{0},and all y ∈ I \{0}. η ξ The commutation condition allows one to iterate the extensions of the maps of a commuting pair. We can also use it to repeat the glueing construction. That is, given a critical commuting pair ζ = (η, ξ) we can regard the interval I = [η(0), ξ ◦ η(0)] as a circle, identifying η(0) and ξ ◦ η(0) and deﬁne f : I → I by η ◦ ξ(x) for x ∈[η(0), 0] f = . η(x) for x ∈[0,ξ ◦ η(0)] HYPERBOLICITY OF RENORMALIZATION 7 FIG. 1. — Acommuting pair The mapping ξ extends to a C -diffeomorphism of open neighborhoods of η(0) and ξ ◦ η(0). Using it as a local chart we turn the interval I into a closed one- dimensional manifold M. Condition (II) above implies that the mapping f projects to a well-deﬁned C -smooth homeomorphism F : M → M. Identifying M with the circle by a diffeomorphism φ : M → T we recover a critical circle mapping φ −1 f = φ◦F ◦φ . The critical circle mappings corresponding to two different choices of φ are conjugated by a diffeomorphism, and thus we recovered a C -smooth conjugacy class of circle mappings from a critical commuting pair. We can metrize the space of C -smooth commuting pairs considered modulo an afﬁne conjugacy as follows (see [dFdM1]). Let ζ = (η ,ξ ), ζ = (η ,ξ ) be 1 1 1 2 2 2 two such pairs, and denote w : C → C aM¨ obius transformation which maps the ordered triple of points η (0), 0, ξ (0) to 0, 1/2, 1.The C -distance between ζ i i 1 and ζ is set to be dist r (ζ ,ζ ) = max |ξ (0)/η (0) − ξ (0)/η (0)|, C 1 2 1 1 2 2 −1 −1 dist r w ◦ ζ ◦ w , w ◦ ζ ◦ w . C 1 1 2 2 1 2 Let f be a critical circle mapping, whose rotation number ρ has a continued fraction expansion (1.1) with at least m+ 1 terms, and let p /q =[r , ..., r ].The m m 0 m−1 q q m+1 m pair of iterates f and f restricted to the circle arcs I and I correspond- m m+1 ingly can be viewed as a critical commuting pair in the following way. Let f be ¯ ¯ the lift of f to the real line satisfying f (0) = 0,and 0< f (0) <1.For each m> 0 let I ⊂ R denote the closed interval adjacent to zero which projects down to the interval I .Let τ : R → R denote the translation x → x + 1.Let η : I → R, m m −p q −p q m+1 m+1 m m ¯ ¯ ¯ ξ : I → R be given by η ≡ τ ◦ f , ξ ≡ τ ◦ f . Then the pair of maps m+1 q q m+1 m ¯ ¯ (η|I ,ξ|I ) forms a critical commuting pair corresponding to ( f |I , f |I ). m m+1 m m+1 Henceforth we shall simply denote this commuting pair by q q m+1 m (2.1) ( f |I , f |I ). m m+1 This allows us to readily identify the dynamics of the above commuting pair with that of the underlying circle map, at the cost of a minor abuse of notation. 8MICHAELYAMPOLSKY Following [dFdM1], we say that the height χ(ζ) of a critical commuting pair ζ = (η, ξ) is equal to r,if r r+1 0 ∈[η (ξ(0)), η (ξ(0))]. If no such r exists, we set χ(ζ) =∞, in this case the map η|I has a ﬁxed point. For a pair ζ with χ(ζ) = r< ∞ one veriﬁes directly that the mappings r r η|[0,η (ξ(0))] and η ◦ ξ|I again form a commuting pair. For a commuting pair ζ = (η, ξ) we will denote by ζ the pair ( η|I , ξ |I ) where tilde means rescaling by η ξ the linear factor λ =− . |I | Deﬁnition 2.2. —The renormalization of a real commuting pair ζ = (η, ξ) is the commuting pair r r Rζ = (η ◦ ξ |I , η|[0,η (ξ(0))]). r r The non-rescaled pair (η ◦ ξ|I ,η|[0,η (ξ(0))]) will be referred to as the pre- renormalization pRζ of the commuting pair ζ = (η, ξ). For a pair ζ we deﬁne its rotation number ρ(ζ) ∈[0, 1] to be equal to the continued fraction [r , r , ...] where r = χ(R ζ). In this deﬁnition 1/∞ is under- 0 1 i stood as 0, hence a rotation number is rational if and only if only ﬁnitely many renormalizations of ζ are deﬁned; if χ(ζ) =∞, ρ(ζ) = 0. Thus deﬁned, the ro- tation number of a commuting pair can be viewed as a rotation number in the usual sense: Proposition 2.1. — The rotation number of the mapping F is equal to ρ(ζ). There is an advantage in deﬁning ρ(ζ) using a sequence of heights in removing the ambiguity in prescribing a continued fraction expansion to rational rotation numbers in a renormalization-natural way. For ρ =[r , r , ...]∈[0, 1] letusset 0 1 G(ρ) =[r , r ,...]= , 1 2 where {x} denotes the fractional part of a real number x (G is usually referred to as the Gauss map). As follows from the deﬁnition, ρ(Rζ) = G(ρ(ζ)) for a real commuting pair ζ with ρ(ζ) = 0. The renormalization of the real commuting pair (2.1), associated to some q q m+2 m+1 critical circle map f ,is the rescaled pair ( f |I , f |I ).Thusfor a given m+1 m+2 critical circle map f the renormalization operator recovers the (rescaled) sequence of the ﬁrst return maps: q q i+1 ( f |I , f |I ) . i i+1 i=1 HYPERBOLICITY OF RENORMALIZATION 9 2.3. Explanation of universality: Lanford’s Program The golden-mean universality phenomenon described above was explained in [ORSS] and [FKS] by a conjectural hyperbolicity property of the operator R. The conjecture was later generalized by various authors, particularly by Lanford [Lan2], to account for more complex universalities. We present it below in its most general form, known as Lanford’s Program. First, a deﬁnition. A critical commuting pair is a commuting pair (η, ξ) whose maps are real-analytic. We shall also impose a technical assumption that ξ analyt- ically extends to an interval (a, b) 0 with ξ(a, b) ⊃[η(0), ξ(0)], and has a single critical point 0 in this interval. The space of critical commuting pairs modulo afﬁne conjugacy will be denoted by C; its subset consisting of pairs ζ with χ(ζ) =∞ will be denoted by S . Renormalization is a transformation R : C \C → C. It ∞ ∞ is not difﬁcult to show (see [Ya2]) that this transformation is injective: Proposition 2.2. —The map R : C \S → C is one-to-one. To outline Lanford’s Program, let us give an informal description of the action of renormalization on the space of critical commuting pairs. −1 Let ρ = φ(θ) : T → T be a monotone continuous function such that φ (ρ) is a point for ρ irrational and an interval otherwise. An example of such a function is the dependence θ → ρ( f ) of the rotation number of a standard map on the parameter. Imagine the space of critical commuting pairs as an inﬁnite-dimensional cylinder C = T ×C , where the rotation number of a commuting pair ζ(θ, ·) with the equatorial coordinate θ ∈ T is ρ(θ). The cylinder C is partitioned into strips (cf. Figure 2) S ={ζ ∈ C|ρ(ζ) =[r, r , ...]} for r = 1, ..., ∞ . r 1 FIG. 2. 10 MICHAEL YAMPOLSKY A boundary component of the strip S is the hypersurface P ⊂ S with the ∞ ∞ ∞ property that a pair ζ ∈ P if and only if it has a single ﬁxed point with unit eigenvalue (ζ is a “parabolic pair”). The sets S accumulate on P in a clockwise r ∞ direction. It is natural to think of the transformation R : C \S → C as being deﬁned piecewise, given on each S , r =∞ by the formula: R : (η, ξ) → (η, η ◦ ξ). The operator R expands the strip S in the equatorial direction, mapping it onto r r a thin equatorial cylinder, which intersects all the vertical strips. We may close the domain of R by adjoining the hypersurface P to it and setting R on P to ∞ ∞ an appropriately deﬁned (multivalued) parabolic renormalization transformation P, with this convention the image of P is also an equatorial cylinder. Lanford con- jectured that this picture can be appropriately endowed by a smooth metric, so that the operator R is uniformly expanding in the θ direction and uniformly con- tracting in the complementary directions. Thus R posesses an inﬁnite-dimensional “horseshoe” structure. The “boxes” R (S ∩ R (S ∩ ··· (R S ) ··· ) r r r r r r −1 −1 −2 −2 −n −n −1 −1 −1 ∩S ∩ R R ··· R S ··· r r 0 n r r r 0 1 n−1 are exponentially small in n, and their intersection is the R-invariant hyperbolic Cantor set A parametrized by the bi-inﬁnite sequences of r ∈ N ∪{∞}.The set A is a “horseshoe” attractor for the renormalization operator, dist(R ζ, A ) → 0 at a geometric rate. The action of R on A is conjugate to the two-sided shift. Let us conclude this exposition by formalizing the Hyperbolicity Conjecture as follows: Renormalization Hyperbolicity Conjecture (Lanford’s Program). — The renormalization operator R in the space of critical commuting pairs possesses a “horseshoe” attractor A on which its action is conjugated to the two-sided shift. Moreover, there exists a renormalization- invariant space of critical commuting pairs with the structure of an inﬁnite dimensional smooth manifold, with respect to which A is a hyperbolic set with one-dimensional expanding direction. If t denotes the expanded coordinate in a local chart, then the dependence t → ρ(t, ·) is continuous and (not strictly) monotone. It is not difﬁcult to see that this Conjecture explains, in particular, the golden mean universality phenomenon we described. Indeed, it implies the existence of a unique ﬁxed point ζ = (η ,ξ ) ∈ A with ρ(ζ ) = ( 5 − 1)/2.The ﬁxed point ∗ ∗ ∗ ∗ ζ is hyperbolic, let us denote its unique expanding eigenvalue δ and set the ∗ ∗ HYPERBOLICITY OF RENORMALIZATION 11 scaling factor |I |/|I |= λ .Then for any ζ with ρ(ζ) = ( 5 − 1)/2 we have ξ η ∗ ∗ ∗ ◦n R ζ → ζ which explains the universality of λ .Moreover, let ζ be a mono- ∗ ∗ µ −n tone one-parameter family of critical commuting pairs. Since the slices (R ) (S ) 1 1 accumulate at the local stable manifold of ζ at the geometric rate 1/δ ,they ∗ ∗ −n have “width” ∼ δ .Providedthe family ζ crosses the local stable manifold of ζ µ ∗ transversely, this implies that the sizes of parameter intervals I ={µ : ζ ∈ S } n µ n as well. decay at the rate 1/δ 3. The passage from smooth to holomorphic The golden-mean universality is observed in any one-parameter family of smooth critical circle maps. We chose to formulate the Hyperbolicity Conjecture for analytic commuting pairs, in part to ensure that the action of renormalization is injective. A deeper reason is the following. In 1986 Eckmann and Epstein [EE] introduced a space of critical commuting pairs now known as the Epstein class. They showed that this class is invariant under the action of R, and constructed the golden-mean ﬁxed point of R in this class using the methods of geometric complex analysis. It was further shown by various people, such as Sullivan (in the unimodal case), Swia¸tek, Herman, and Yoccoz, that the renormalizations of any C -smooth commuting pair with an irrational rotation number converge to the Epstein class, at a geometric rate in the C -metric. Below, after some preliminaries, we deﬁne the Epstein class, and formulate these results more precisely. 3.1. Carathe´odory topology on a space of branched coverings Consider the collection X of all triplets (U, u, f ),where U ⊂ C is a topo- logical disk different from the whole plane, u ∈ U,and f : U → C is a three-fold analytic branched covering map, with the only branch point at u. We will topol- ogize X as follows (cf. [McM1]). Let {(U , u )} be a sequence of open connected regions U ⊂ C with marked n n n points u ∈ U . Recall that this sequence Caratheodory converges to a marked region n n (U, u) if: • u → u ∈ U,and • for any Hausdorff limit point K of the sequence C \ U , U is a component of C \ K. For a simply connected U ⊂ C and u ∈ U let R : D → U denote the inverse (U,u) Riemann mapping with normalization R (0) = u, R (0) >0. By a classical (U,u) (U,u) result of Caratheodory, the Carathedory convergence of simply-connected regions ´ ´ 12 MICHAEL YAMPOLSKY (U , u ) → (U, u) is equivalent to the locally uniform convergence of the inverse n n Riemann mappings R to R . (U ,u ) (U,u) n n For positive numbers , , and compact subsets K and K of the open 1 2 3 1 2 unit disk D, let the neighborhood U (U, u, f ) of an element (U, u, f ) ∈ X , , ,K ,K 1 2 3 1 2 be the set of all (V, v, g) ∈ X,for which: •|u − v| < , • sup |R (z) − R (z)| < , (V,v) (U,u) 2 z∈K • and R (K ) ⊂ V,and sup |f (z) − g(z)| < . (U,u) 2 3 z∈R (K ) (U,u) 2 One veriﬁes that the sets U (U, u, f ) form a base of a topology on X, , , ,K ,K 1 2 3 1 2 which we will call Caratheodory topology. This topology is clearly Hausdorff, and the convergence of a sequence (U , u , f ) to (U, u, f ) is equivalent to the Caratheodory n n n convergence of the marked regions (U , u ) → (U, u) as well as a locally uniform n n convergence f → f . 3.2. The Epstein class An orientation preserving interval homeomorphism g : I =[0, a]→ g(I) = J belongs to the Epstein class E if it extends to an analytic three-fold branched cov- ering map of a topological disk G ⊃ I onto the double-slit plane C ,where J ⊃ cl J. Any map g in the Epstein class can be decomposed as (3.1) g = Q ◦ h, where Qc(z) = z + c,and h : I →[0, b] is a univalent map h : G → ∆ (h) onto the complex plane with six slits, which triple covers C under the cubic map Qc(z). For any s ∈ (0, 1), let us introduce a smaller class E ⊂ E of Epstein map- −1 pings g : I =[0, a]→ J ⊂ J for which both |I| and dist(I, J) are s -commensurable with | J|, the length of each component of J \ J is at least s| J|,and g (a) >s.We will often refer to the space E as the Epstein class, and to each E as an Epstein class. C belongs to the (an) Epstein class if We say that a commuting pair (η, ξ) ∈ both of its maps do. Similarly, a critical circle map f is Epstein if Rf is in the Epstein class. It immediately follows from the deﬁnitions that: Lemma 3.1. — If a renormalizable commuting pair ζ is in the Epstein class, then the same is true for Rζ . Let us make a note of an important compactness property of E s HYPERBOLICITY OF RENORMALIZATION 13 Lemma 3.2 (Lemma 2.10 [Ya2]). — Let s ∈ (0, 1). The collection of normalized maps g ∈ E with I =[0, 1], with marked domains (U, 0) is sequentially compact with respect to Caratheodory topology. 3.3. Real a priori bounds An excellent exposition of the results of Herman, Swia¸tek, and Yoccoz on the real geometry of critical circle maps is found in [dFdM1]. We formulate some of the main results. Real a priori bounds of Herman and Swia¸tek. — There exists a universal constant K such that the following holds. Let f be a critical circle map. Denote Π the partition of the circle by the iterates q −1 q −1 q −1 q −1 q i m−1 q i m q +q i m−1 q +q i m m m−1 m−1 m m m−1 { f (0)} ∪{ f (0)} ∪{ f (0)} ∪{ f (0)} i=0 i=0 i=0 i=0 (note that I and I are two adjacent intervals of the partition). Then there exists M = m m−1 M( f ) such that for every m>M any two adjacent intervals of the partition are K- 1 m ∞ commensurable. Moreover, the C -norm of the renormalizations {R ( f )} is bounded m=M by K. Utilizing the above bounds Herman has shown in [He] (see [dFdM1] for a published account): Theorem 3.3. — Any two critical circle maps with the same irrational rotation numbers are quasisymmetrically conjugate. The importance of the Epstein class lies in the fact that all C -limit points m ∞ of the sequence {R ( f )} are in E for a universal value of s> 0.A more m=M precise formulation of this was proved in the recent work of de Faria and de Melo [dFdM1]: Lemma 3.4. — There exists a universal constant s> 0 such that the following holds. Let f ∈ C , (r ≥ 3) be a critical circle map with an irrational rotation number. Then the m r−1 q q m+1 sequence of real commuting pairs R ( f ) = ( f |I , f |I ) is bounded in C -metric, m m+1 r−1 and C -converges to E at a geometric rate. In particular, for a critical circle map f ∈ E there exists σ >0 such that all its renormalizations are contained in E . Moreover, the constant σ can be chosen independent on f , after skipping the ﬁrst few renormalizations. Finally, let us formulate a technical statement about Epstein pairs to be used further in the paper: 14 MICHAEL YAMPOLSKY Lemma 3.5 (Lemma 2.13, [Ya2]). — Let ζ = (η, ξ) ∈ E be a critical commuting pair with ρ(ζ) = 0, which appears as a limit of a sequence {ζ }⊂ E with ρ(ζ ) ∈ R \ Q . n n Then the map η has a unique ﬁxed point in the interval I , which is necessarily parabolic, with multiplier one. Acommuting pair ζ = (η, ξ) ∈ E will be called parabolic if the map η has a unique ﬁxed point in I , which has a unit multiplier; this point will usually be denoted p . Note, that by virtue of its uniqueness, p has to be globally attracting η η on one side for the interval homeomorphism η| , it is globally attracting on the −1 other side under η . 4. Outline of the results 4.1. Known results The early efforts to prove the Renormalization Hyperbolicity Conjecture cul- minated in an argument of Mestel [Mes], who established the existence and local hyperbolicity properties of a ﬁxed point of R in C . The argument was based on rigorous computer estimates. This approach was developed by Lanford in the context of the renormalization of unimodal maps. One of its major drawbacks is its local nature; neither the uniqueness of the ﬁxed point, nor global hyperbolicity of R are established. In his address to ICM-86 in Berkeley Sullivan [Sul1] outlined a program of constructing the invariant set of renormalization of unimodal maps and its stable set by means of the Teichmul¨ ler theory, which he subsequently carried out (see [Sul2],[MvS]). Sullivan’s program has been adapted to the setting of critical circle mappings by de Faria [dF1],[dF2]. De Faria has deﬁned, in particular, extensions of Epstein commuting pairs to holomorphic dynamical systems in the plane, which are analogous to quadratic-like mappings in the unimodal renormalization theory. We discuss these extensions, called holomorphic commuting pairs,in §5. Using Sullivan’s Riemann surface laminations machinery, de Faria constructed a horseshoe attractor A for the renormalization operator acting on the space of Epstein commuting pairs of type bounded by some constant B (this corresponds to restricting the operator R to the ﬁnite collection of vertical strips ∪S , r ≤ B in Figure 2). In a recent paper [Ya2] we generalized the results of de Faria to Epstein commuting pairs of unbounded combinatorial type. Denote Σ the space of bi-inﬁnite sequences (..., r , ..., r , r , r , ..., r , ...) with r ∈ N ∪{∞} equipped with the weak topology. −k −1 0 1 k i Our theorem completely settles the ﬁrst part of Lanford’s Program: Theorem 4.1 (Renormalization horseshoe). — There exists an R-invariant closed set A ⊂ E consisting of pairs with irrational rotation numbers and parabolic pairs with the HYPERBOLICITY OF RENORMALIZATION 15 following properties. The action of R on A is topologically conjugate to the two-sided shift ¯ ¯ σ : Σ → Σ: −1 i ◦ R ◦ i = σ −1 and if ζ = i (..., r , ..., r , r , r , ..., r , ...) then ρ(ζ) =[r , r , ..., r , ...].For any −k −1 0 1 k 0 1 k ζ ∈ E with irrational rotation number we have R ζ → A in the Caratheodory topology. Moreover, for any two pairs ζ , ζ ∈ E with ρ(ζ) = ρ(ζ ) we have n n dist(R ζ, R ζ ) → 0 for the uniform distance between analytic extensions of the renormalized pairs on compact sets. The proof of Theorem 4.1 combines the ideas of McMullen from [McM2] with apriori bounds obtained in [Ya1]. We consider the geometric limits of the rescaled sequences of renormalizations ζ, Rζ, ..., R ζ. These limits which we call bi-inﬁnite limiting renormalization towers (see §5for the deﬁnition) should be thought of as bi-inﬁnite sequences of nested holomorphic commuting pairs in the plane. Each tower corresponds to a single point in the attractor A , and the result follows from a Rigidity Theorem [Ya2]: Theorem 4.2 (Tower rigidity). — Any two towers with the same rotation numbers are afﬁnely conjugated. As far as the hyperbolicity aspects of Lanford’s Program, the following is known. In the case of a bounded type rotation number, de Faria and de Melo [dFdM2] strengthened the original result of de Faria, utilizing a technique of Mc- Mullen [McM2] to show that the rate of converegence to A is geometric: Theorem 4.3 ([dFdM2]). — For every B ∈ N and every r ∈ N ∪ 0 there exists α <1 such that for every ζ ∈ E with ρ(ζ) ∈ R \ Q of type bounded by B and for every r,B ˆ ˆ ζ ∈ A with ρ(ζ) = ρ(ζ) we have n n n dist (R ζ, R ζ) < const ·(α ) . C r,B Despite the difﬁculty in imposing a manifold structure on the space of com- muting pairs compatible with the analyticity of R, it has long been known how one may construct an unstable direction in such a manifold. A version of this construction was ﬁrst shown to us by Folkert Tangerman, who participated in its discussion in Sullivan’s seminar at the Graduate Center of CUNY. We give a rather simple argument in §9. 16 MICHAEL YAMPOLSKY 4.2. Statement of the main result First, some deﬁnitions. Suppose, B is a complex Banach space whose elem- ents are functions of the complex variable. Let us say that the real slice of B is the real Banach space B consisting of the real-symmetric elements of B.If X is a Banach manifold modelled on B with the atlas {Ψ } we shall say that X is −1 R R real-symmetric if Ψ ◦Ψ (B ) ⊂ B for any pair of indices γ , γ .The real slice of X γ 1 2 1 γ R −1 R is then deﬁned as the real Banach manifold X ⊂ X given by Ψ (B ) in a local R R chart Ψ .Anoperator A deﬁned on a subset of X is real-symmetric if A(X ) ⊂ X . A particular example we shall introduce in §7 is a real-analytic manifold C of critical cylinder maps. Its real slice M ≡ C consists of critical circle maps deﬁned in an annulus U ⊃ T in the cylinder C/Z. We will construct a closed equivalence relation denoted ∼ between com- conf muting pairs in the Epstein class E (with s being as in Lemma 3.4) with the property that if ζ ∼ ζ , then the analytic extensions of the commuting pairs are 1 conf 2 conjugate in a speciﬁc neighborhood of the interval of deﬁnition by a conformal change of coordinates. In particular, ρ(ζ ) = ρ(ζ ). Moreover, there exists a value 1 2 N N N ∈ N such that if ζ and ζ are N-times renormalizable, then R ζ ∼ R ζ . 1 2 1 conf 2 This last property implies that periodic orbits of R constructed in Theorem 4.1 project to the quotient space. The main result of this paper is the Renormalization Hyperbolicity Theorem in the following form: Theorem 4.4 (Renormalization Hyperbolicity). — There exists a real-symmetric analytic operator R from an open subset of C to C (thus R maps an open subset of M cyl U U cyl into M), such that the following holds. For every r =∞ there is a canonical homeomorphism onto the image S ∩ E / ∼ −→ M r s conf N −1 such that ρ(ι (ζ)) = ρ(R(ζ)),and ι ◦ R ◦ ι ≡ R . For every periodic point p ∈ S r r cyl r of the operator R there exists ∈ N such that the point ι ( p) is a hyperbolic ﬁxed point of R | , with a one-dimensional unstable direction. cyl Note, in particular, that we have found a way to deﬁne a renormalization for critical circle maps themselves, without recourse to critical commuting pairs. Our theorem implies the existence of universal constants associated to every rotation number of a periodic type, and in particular we get a version of the golden-mean universality: Corollary 4.5 (Golden-Mean Universality). — There exists a pair of universal constants δ >1 and λ >0, such that the following holds. Let { f } be a one-parameter family of analytic µ HYPERBOLICITY OF RENORMALIZATION 17 critical circle maps with a C -smooth dependence on the parameter µ.Denote f : R → R a continuously chosen lift of f : T → T, and assume additionally that ∂f (x)/∂µ >0 for µ µ any x ∈ R (the Arnold’s family is one example of such family). Suppose ρ( f ) = ( 5−1)/2 and denote I the closed interval of values of the parameter for which ρ( f ) = p /q where n µ n n p /q is the n-th convergent of the golden mean. Then: n n lM •|I |/δ → a> 0 for some M ∈ N; k+lM q q n n+1 •| f (0)|/| f (0)|→ λ geometrically fast. µ µ ∗ ∗ We remark that the second statement already follows from [dFdM2]. 5. Holomorphic commuting pairs 5.1. Deﬁnitions De Faria [dF1,dF2] introduced holomorphic commuting pairs to apply Sul- livan’s Riemann surface laminations technique to the renormalization of critical circle maps. They are suitably deﬁned holomorphic extensions of critical commut- ing pairs which replace Douady-Hubbard polynomial-like maps [DH2]. A critical commuting pair ζ = (η| ,ξ| ) extends to a holomorphic commuting pair H if there I I η ξ exist four simply-connected R-symmetric domains ∆ , D, U, V such that ¯ ¯ ¯ ¯ ¯ • D, U, V ⊂ ∆ , U ∩ V ={0}; the sets U \ D, V \ D, D \ U,and D \ V are nonempty, connected, and simply-connected, U ∩ R = I , V ∩ R = I ; η ξ • mappings η : U → (∆ \ R) ∪ η(I ) and ξ : V → (∆ \ R) ∪ ξ(I ) are onto η ξ and univalent; • ν ≡ η ◦ ξ : D → (∆ \ R) ∪ ν(I ) is a three-fold branched covering with a unique critical point at zero, where I = D ∩ R. We shall call ζ the commuting pair underlying H , and write ζ ≡ ζ .The domain D ∪ U ∪ V of a holomorphic commuting pair H will be denoted Ω or Ω ,the FIG. 3. — A holomorphic commuting pair 18 MICHAEL YAMPOLSKY rangewill bedenoted ∆ or ∆ . The closure of the set of points whose orbit under H is contained in Ω will be referred to as the ﬁlled Julia set of H , denoted K(H ).The Julia set of H is deﬁned as J(H ) = ∂K(H ). It is easy to see directly from the deﬁnition (cf. [dF2]) that: Proposition 5.1. —Let ζ be a commuting pair with χ(ζ) < ∞. Suppose ζ is a re- striction of a holomorphic commuting pair H , that is ζ = ζ . Then there exists a holomorphic commuting pair G with range ∆ , such that ζ = Rζ . H G The shadow of the holomorphic commuting pair H is the following piecewise- deﬁned holomorphic dynamical system: η(z), z ∈ U S (z) = ξ(z), z ∈ V ξ ◦ η(z), z ∈ D \ (U ∪ V). As the next proposition shows one may think of the shadow of a holomorphic commuting pair as an analogue of a cubic-like map: Proposition 5.2 (Prop. II.4. [dF2]). — Given a holomorphic commuting pair H as −1 above, consider its shadow S .Let I = Ω ∩ R,and X = I ∪ S (I).Then: • The restriction of S to Ω \ X is a regular three fold covering onto ∆ \ R. • S and H share the same orbits as sets. We will say that two holomorphic commuting pairs H : Ω → ∆ and H H G : Ω → ∆ are conjugate if there is a homeomorphism h : ∆ → ∆ such G G G H that −1 S = h ◦ S ◦ h. G H −1 In this case we will simply write G = h ◦ H ◦ h. 5.2. Complex a priori bounds We shall denote by H the space of holomorphic commuting pairs H : Ω → ∆ whose underlying real commuting pair (η, ξ) is in the Epstein class. In this case both maps η and ξ extend to triple branched coverings η ˆ : U → ∆ ∩ C η( J ) ˆ ˆ and ξ : V → ∆ ∩ C respectively. We will turn H into a topological space by ξ( J ) ˆ ˆ ˆ identifying it with a subset of X ×X by H → (U, 0, η) × (V, 0, ξ) (cf. §3.1). We say that a real commuting pair (η, ξ) with an irrational rotation num- ber has complex apriori bounds, if all its renormalizations extend to holomorphic commuting pairs with bounded moduli: mod(∆ \ Ω ) > µ >0. HYPERBOLICITY OF RENORMALIZATION 19 For µ ∈ (0, 1) let H(µ) denote the space of holomorphic commuting pairs H : Ω → ∆ ,with mod(∆ \ Ω ) > µ, min(|I |, |I |) > µ and diam(∆ ) <1/µ. H H H H η ξ H Lemma 5.3 (Lemma 2.15 [Ya2]). — For each µ ∈ (0, 1) the space H(µ) is sequen- tially pre-compact, with every limit point contained in H(µ/2). The existense of complex apriori bounds is a key analytic issue of renor- malization theory. In the case of critical circle maps it is settled by the following theorem: Theorem 5.4. — There exist universal constants µ >0 and K> 1 such that the following holds. Let ζ ∈ C be a critical commuting pair with an irrational rotation number. Then there exists N = N(ζ) such that for all n ≥ N the commuting pair R ζ extends to a holomorphic commuting pair H : Ω → ∆ in H(µ).The range ∆ is a Euclidean disk, n n n n and the regions Ω ∩ (±H) are K-quasidisks. Remark 5.1. — We ﬁrst proved this theorem in [Ya1] for commuting pairs ζ in an Epstein class E ,inwhich case N = N(s). Our proof was later adapted by de Faria and de Melo [dFdM2] to the case of a non-Epstein critical commuting pair. In the general case, in a Carath´ eodory compact family of critical commuting pairs, the number N canbechosenuniformly. Let ζ be at least n times renormalizable critical commuting pair. For the lack of a better term, let us say that the pair of numbers τ (ζ) = (r , r ) forms n n−1 n−2 the history of the pair R ζ . Based on the above theorem and a detailed analysis of the shapes of the domains Ω we proved the following in [Ya1]: Theorem 5.5 ([Ya1]). — There exists a universal constant K >1 such that the follow- ing holds. Let ζ = (η ,ξ ) and ζ = (η ,ξ ) be two critical commuting pairs with irrational 1 1 1 2 2 2 rotation numbers. Let n> max(N(ζ ), N(ζ )) + 1 as above. Assume that the n-th renormal- 1 2 izations of ζ , ζ have the same rotation number and the same history. Then their holomorphic 1 2 1 2 commuting pair extensions H , H are K −quasiconformally conjugate. The conjugating map n n is conformal on the ﬁlled Julia set. For commuting pairs of the type bounded by B this theorem was ﬁrst proved by de Faria [dF1,dF2], with “K ” depending on the value of B. 5.3. Limiting renormalization towers Sullivan’s original proof of the existence and uniqueness of the Feigenbaum ﬁxed point used the Teichmuller theory for Riemann surface laminations asso- ciated to quadratic-like maps, which he himself developed. McMullen [McM2] 20 MICHAEL YAMPOLSKY has replaced this approach with an elegant argument based on a rigidity result for dynamical objects he called renormalization towers. The renormalization tow- ers of critical commuting pairs were studied in [dFdM2] for pairs of bounded type, and later in [Ya2] without the restriction on the type. Their deﬁnition is as follows. Let {H } be a sequence of generalized holomorphic commuting pairs. Set ζ = ζ . Suppose m(k) >n(k) are two sequences of natural numbers such k H that n(k) →∞, m(k) − n(k) →∞ and ζ is m(k)-times renormalizable. Moreover, n(k)+i i the pre-renormalizations pR ζ have holomorphic pair extensions H such that i ∞ H → H ∈ H,for i ∈ Z. Then the bi-inﬁnite sequence T = (H ) is referred i i k −∞ to as a bi-inﬁnite limiting renormalization tower.The rotation number ρ(T ) is naturally deﬁned as the bi-inﬁnite sequence of heights (χ(ζ )) . The precise formulation i −∞ of Theorem 4.2 is: Tower Rigidity Theorem ([Ya2]). — For any bi-inﬁnite sequence ρ ¯ = (χ ) ,χ ∈ n n −∞ N∪{∞} there exists a bi-inﬁnite limiting renormalization tower T with ρ(T ) = ρ ¯.Moreover, 1 ∞ 2 ∞ for any two limiting renormalization towers T = (H ) , T = (H ) with ρ(T ) = 1 2 1 i −∞ i −∞ ρ(T ) we have ζ 1 ≡ ζ 2 for all i. H H i i 6. Parabolic mapsand theirperturbations General facts. — We begin with a brief review of the theory of parabolic bifurcations, as applied in particular to an interval map in the Epstein class. For a more comprehensive exposition the reader is referred to [Do], supporting tech- nical details may be found in [Sh]. Fix a map η ∈ E having a parabolic ﬁxed point p with unit multiplier. A R Theorem 6.1 (Fatou Coordinates). — There exist topological discs U and U , called attracting and repelling petals, whose union is a punctured neighborhood of the parabolic periodic point p such that A A k A ¯ ¯ η (U ) ⊂ U { p}, and η (U ) ={ p}, k=0 R R −k R ¯ ¯ η (U ) ⊂ U { p}, and η (U ) ={ p}, k=0 −1 where η is the univalent branch ﬁxing ζ . Moreover, there exist injective analytic maps A A R R Φ : U → C and Φ : U → C, HYPERBOLICITY OF RENORMALIZATION 21 unique up to post-composition by translations, such that A A R R Φ (η (z)) = Φ (z) + 1 and Φ (η (z)) = Φ (z) − 1. 0 0 A A R R The Riemann surfaces C = U /η and C = U /η are conformally equivalent to the 0 0 cylinder C/Z. Proof. — To ﬁx our ideas, let us assume that p>0. We will show that p is a simple parabolic point of η ,thatis, 2 2 η (z − p) = (z − p) + a(z − p) + o((z − p) ), with a = 0. Indeed, for a small c<0 consider the map η = η +c.As η ∈ E,there c 0 c exists a well-deﬁned inverse branch φ of this map in C \ R which preserves both the upper and the lower half-planes. Since η has no real ﬁxed points, the Denjoy- Wolff Theorem implies that there exists a pair of complex conjugate ﬁxed points ± + p ∈±H such that every orbit originating in H converges to p and similarly c c for the other ﬁxed point. Thus, p splits into a pair of repelling ﬁxed points, and hence its algebraic multiplicity is two. The rest of the construction is classical. A A R R We denote π : U → C and π : U → C the natural projections. The A R A R quotients C and C are customarily referred to as Ecalle-Voronin cylinders;we will ﬁnd it useful to regard these as Riemann spheres with distinguished points +, − A A ﬁlling in the punctures. The real axis projects to the natural equators E ⊂ C and R R A R E ⊂ C .Any conformal transit homeomorphism τ : C → C ﬁxing the ends +, − is a translation in suitable coordinates. Liﬁting it produces a map τ : U → C satisfying τ ◦ π = π ◦ τ. A R We will sometimes write τ ≡ τ ,and τ ¯ = τ ¯ ,where θ θ R A −1 Φ ◦ τ ◦ (Φ ) (z) ≡ z + θ mod Z. Suppose for an Epstein map η in a sufﬁciently small neighborhood of η the parabolic point splits into a complex conjugate pair of repelling ﬁxed points p ∈ H ± 2π i±α(η) and p ¯ with multipliers λ = e . In this situation one may still speak of attracting and repelling petals: Lemma 6.2 (Douady Coordinates). — There exists a neighborhood U(η ) ⊂ E of the map η such that for any η ∈ U(η ) with | arg α(η)| < π/4, there exist topological discs 0 0 A R U and U whose union is a neighborhood of p, and injective analytic maps η η A A R R Φ : U → C and Φ : U → C η η f 22 MICHAEL YAMPOLSKY unique up to post-composition by translations, such that A A R R Φ (η(z)) = Φ (z) + 1 and Φ (η(z)) = Φ (z) + 1. η η η η A A R R The quotients C = U /η and C = U /η are Riemann surfaces conformally equivalent to η η η η C/Z. Let us note: Proposition 6.3. — There exists an open neighborhood W (η ) ⊂ E of η in the 0 s 0 Caratheodory topology, such that for every η ∈ W (η ) as above, the condition on the eigenvalues of the repelling ﬁxed points is automatically satisﬁed. Proof. — Recall that if z is an isolated ﬁxed point of an analytic map f , then the holomorphic index of z is the residue 1 dz i( f , z ) = 2π i z − f (z) where the integrating is done over a small contour enclosing z (see [Mil]). In the case when λ = f (z) = 1,anelementary computationshows that i( f , z ) = 1/(1 − λ).In our case,bycontinuity, 1 1 + −→ i(η , p). + − η→η 1 − λ 1 − λ 0 η η + ± Since the right-hand side is ﬁnite, and λ = λ ,and 1 − λ −→ 0, it follows that η η η η→η arg 1 − λ −→ . η→η 0 2 A R An arbitrary choice of real basepoints a ∈ U and r ∈ U enables us to specify the Fatou and Douady coordinates uniquely, by requiring that Φ (a) = A R R Φ (a) = 0,and Φ (r) = Φ (r) = 0. The following fundamental theorem ﬁrst η η appeared in [DH1]: A R Theorem 6.4. — With these normalizations the maps Φ , Φ depend continuously on η η η with respect to the compact-open topology, and A A R R Φ → Φ and Φ → Φ η η A R uniformly on compact subsets of U and U respectively. n(η) Moreover, select the smallest n(η) ∈ N for which η (a) ≥ r.Then −1 n(η) R A η (z) = Φ ◦ T ◦ Φ θ(η)+K η η HYPERBOLICITY OF RENORMALIZATION 23 wherever both sides are deﬁned. In this formula T (z) denotes the translation z → z + a, θ(η) ∈[0, 1) is given by θ(η) = 1/α(η) + o(1) mod 1, α(η)→∞ and the real constant K is determined by the choice of the basepoints a, r. Thus for a sequence n(η ) {η }⊂ U(η) converging to η, the iterates η converge locally uniformly if and only if there is a convergence θ(η) → θ , and the limit in this case is a certain lift of the transit homeomorphism τ for the parabolic map η . θ 0 As a ﬁnal remark in this section, let us note that, for example, Shishikura states Lemma 6.2 and Theorem 6.4 [Sh] for an open set of maps in a Banach manifold, and the proofs he gives actually imply that Douady’s coordinates vary locally analytically with the map η. We will prove a similar statement further in this paper. 7. Renormalization of maps of the cylinder 7.1. A space of maps of the cylinder and a renormalization transformation In this section we will introduce a new renormalization procedure deﬁned on a space of analytic maps of the cylinder. As it will turn out this renormal- ization is naturally related to the renormalization of critical circle maps, and has some important advantages over the latter. Firstly, let us denote π the natural projection C → C/Z. For an equatorial annulus U ⊂ C/Z let A be the space of bounded analytic maps φ : U → C/Z,such that φ(T) is homotopic to T, equipped with the uniform metric. We shall turn A into a real-symmetric com- −1 plex Banach manifold as follows. Denote U the lift π (U) ⊂ C.The space of ˜ ˜ functions φ : U → C which are analytic, continuous up to the boundary, and ˜ ˜ 1-periodic, φ(z + 1) = φ(z), becomes a Banach space when endowed with the sup ˜ ˇ norm. Denote that space A .For a function φ : U → C/Z denote φ an arbitrarily −1 ˇ ˜ ˇ ˜ chosen lift π(φ(π (z))) = φ.Observe that φ ∈ A if and only if φ = φ − Id ∈ A . U U We use the local homeomorphism between A and A given by U U −1 ˜ ˜ φ → π ◦ (φ + Id) ◦ π to deﬁne the atlas on A . The coordinate change transformations are given by ˜ ˜ φ(z) → φ(z + n) + m for n, m ∈ Z, therefore with this atlas A is a real-symmetric complex Banach manifold. Now let f : U → C/Z be an analytic critical circle map. By deﬁnition, there is a neighborhood of the equator in which 0 is the only critical point of f .Let ˜ ˜ f : U → C be a lift of f . The Argument Principle implies that for >0 small 24 MICHAEL YAMPOLSKY enough, if g˜ ∈ A is real-analytic, with g˜ (0) =−1, g˜ (0) = 0,and || f − g˜|| < , −1 then g = π ◦(g˜+Id)◦π is a critical circle map as well. Let ( f ) be the supremum of such ,and set −1 C =∪ π ◦ (g˜ + Id) ◦ π | g˜ ∈ A , g˜ (0) =−1, g˜ (0) = 0, U f U and ||g˜ − f || < ( f ) . Proposition 7.1. — The space C is a codimension 2 submanifold of A . U U Proof. — Set ˜ ˜ ˜ ˜ B ={φ ∈ A with φ (0) = 0, φ (0) = 0}. U U This is a Banach subspace of A , which has codimension 2 by the Implicit Func- tion Theorem. Select an element ψ ∈ A with ψ (0) =−1, ψ (0) = 0.In the local chart on A deﬁned by −1 ˜ ˜ φ → π ◦ (φ + Id) ◦ π the image of C is the afﬁne subspace ψ +B . U U We shall say that f is a critical cylinder map if f ∈ C for some U ⊂ C/Z. Let us denote by C the real slice of C : C ={ f ∈ C with f (z) = f (z)} ={ f ∈ A , such that f | is a critical circle map}. U T Deﬁnition 7.1. — Given a critical cylinder map f ∈ C let us say that it is cylinder renormalizable, or simply renormalizable, if there exists k>1 and an equatorial annulus V ⊂ C/Z such that following holds: • there exist repelling periodic points p , p of f in U with periods k and a simple arc 1 2 k k l connecting them such that f (l) is a simple arc, and f (l) ∩ l ={ p , p }; 1 2 k k • the iterate f is deﬁned and univalent in the domain C bounded by l and f (l),the −k corresponding inverse branch f | univalently extends to C ; and the quotient of f (C ) f C ∪ f (C )\{ p , p } by the action of f is a Riemann surface conformally isomorphic f f 1 2 to the cylinder C/Z (we will call a domain C with these properties a fundamental crescent of f ); j n(z) ¯ ¯ • for a point z ∈ C with { f (z)} ∩ C =∅,set R (z) = f (z) where f j∈N f C n(z) n(z) ∈ N is the smallest value for which f (z) ∈ C . We further require that there exists a point c in the domain of R such that f (c) = 0 for some m<n(c);and ˆ ˆ if we denote f the projection of R to C/Z with c → 0,then f ∈ C . C V We will say that the new critical circle map f is a cylinder renormalization of f with period k. HYPERBOLICITY OF RENORMALIZATION 25 The relation of the cylinder renormalization procedure to critical circle maps is easy to see: Proposition 7.2. — Suppose f ∈ C is a critical circle map with rotation number ρ( f ) ∈ R\Q . Assume that it is cylinder renormalizable with period q . Then the corresponding ˆ ˆ renormalization f is also a critical circle map with rotation number G (ρ( f )).Also, Rf is n+1 analytically conjugate to R f . Proof. — As follows from the deﬁnition of a fundamental crescent, the in- tersection of the domain C with T is an arc of the form [a, f (a)].The claim follows in an obvious fashion from this observation. The ﬁrst useful property of cylinder renormalization is the following: Proposition 7.3. —Let f ∈ C be renormalizable, with a cylinder renormalization f ∈ C corresponding to the fundamental crescent C ,and let W be any equatorial annulus V f compactly contained in V. Then there is an open neighborhood G ⊂ C of f such that every g ∈ G is renormalizable, with a fundamental crescent C ⊂ U which depends continuously on g in the Hausdorff sense. Moreover, there exists a holomorphic motion χ : ∂C → ∂C over G, g f g such that χ ( f (z)) = g(χ (z)). And ﬁnally, the renormalization gˆ is contained in C . g g W Proof. — Firstly, there is a neighborhood of f in which there exist continuous g g f branches of g -periodic points p , p with p = p ,suchthatbothpointsare re- 1 2 i pelling. Shrinking the neighborhood, if necessary, we may select an analytic family g g of linearizing coordinates w : D → D which conjugate the iterate g to the linear i i g g g g g map z → λ z,with λ = (g ) ( p ) (here D is a topological disk around p ,and i i i i i g g g g w ( p ) = 0). We may further assume that w (D ) = D. Suppose C is bounded by i i i i the curves l , f (l ) terminating at the points p ,and let usdenote z the points f f i k k in ∂D ∩ f (l ) which are obtained last, when moving along f (l ) starting from f f −k the equator, and ζ their preimages by the branches of f deﬁned in D and f f f f ﬁxing p . Note that removing a smooth simple curve γ ⊂ D connecting p to i i i i f f the boundary of D from D we may select a branch of i i f g W = log w i i log λ conjugating f to z → z + 1 in the complementary domain. Let us embed γ g g into an analytic family of curves γ with the same properties, and deﬁne W in i i a similar fashion, selecting the same branch of log for all maps in a neighborhood of f . Now we are prepared to deﬁne C .Let l be the segment of l terminating g f at the points ζ , ζ ,and l the complementary segment containing p .Now let 1 2 i i 26 MICHAEL YAMPOLSKY f g L be the curve l in W -coordinates, and L the parallel translate of this curve i i i i g g terminating at W (ζ ).Weset l to be the union of three curves: l,and l = i g i i g g −1 k (W ) (L ),and deﬁne l = g (l ). By construction, shrinking the neighborhood i i g around f again, if necessary, we may ensure that the curves l , l are disjoint away from the ends. Denote C the domain bounded by l , l , chosen so that g g g → C is Hausdorff continuous. By the Grotzsch inequality, C is a fundamental g g crescent for g. The existense of the desired holomorphic motion is obvious from the construction. It remains to show that G may be chosen so that g ∈ C . It clearly follows −k from Deﬁnition 7.1 that there is a domain D ⊂ C ∪ f (C ) and an iterate n> k f f such that W π (D) (where π : C → C/Z denotes the appropriate projection) f f f n n and f ∈ C is the projection of the iterate f | . Now the iterate g | projects to W D D an analytic map g which is in C for all g sufﬁciently close to f . The next proposition shows that g → g is well-deﬁned: Proposition 7.4. — In the notations of of Proposition 7.3, let C be a different family of fundamental crescents, which also depends continuously on g ∈ G in the Hausdorff sense. Then there exists an open neighborhood G ⊂ G of f , such that the cylinder renormalization of g ∈ G corresponding to C is also gˆ. Proof. — The argument is standard in the study of Douady coordinates; we outline it, and leave the details to the reader. As we have seen above, in a slit neighborhood of p ,the iterate g is conjugate to z → z + 1,and both C and C i g become fundamental half-inﬁnite strips for the orbits of the unit translation. Let us deﬁne a projection π : C → C as follows: in a small neighborhood of the re- pelling point p the point z ∈ C will project to a point z ∈ C , which is its integer i g translate in the log-linearizing coordinate. Outside the linearizing neighborhoods of k k −1 the repelling points, every z ∈ C belongs to a union C ∪ f (C ) ∪ ( f | ) (C ) g g g C g (provided G is chosen small enough). We then set π(z) to be one of the points k k −1 z, f (z), ( f | ) (z) which belongs to C . In the quotient cylinders, π becomes g g aconformal map C/Z → C/Z which conjugates the corresponding cylinder renor- malizations of g. However, it also ﬁxes 0 ∈ C/Z and hence is the identity. We invite the reader to verify, along the same lines, that if t → C is a ho- motopy of fundamental crescents for f , then the corresponding cylinder renormal- izations all coincide. However, a critical cylinder map may be renormalizable in several different ways. Further in this section we will make a canonical choice of cylinder renormalization for a particular class of critical cylinder maps. A key property of the cylinder renormalization is the following: HYPERBOLICITY OF RENORMALIZATION 27 Proposition 7.5. — In the notations of Proposition 7.3, the dependence g → g is an analytic map G → C . Proof. — By the Theorem of Bers and Royden [BR], the holomorphic motion k k χ extends to a holomorphic motion χ : C ∪ f (C ) → C ∪ g (C ) over a smaller g g f f g g open neighborhood G with the same equivariance property. As shown in [MSS], the holomorphic motion induces an analytic family of quasiconformal maps k k k k Ψ : C/Z = (C ∪ f (C ))/f → C/Z = (C ∪ g (C ))/g . g f f g g Applying the theorem of Ahlfors and Bers, we see that the projection π : C ∪ g g k −k g (C ) → C/Z depends analytically on g.Let D ⊂ C ∪ f (C ) and n ∈ N be such g f f n n that W π (D),and f ∈ C is the projection of the iterate f | .The iterate g | f W D D projects to π ◦ g = gˆ ∈ C for all g sufﬁciently close to f , and the claim follows. g W Let us summarize. The cylinder renormalization acts on critical circle maps themselves, rather than on commuting pairs. The reason for introducing commut- ing pairs in the ﬁrst place, was the absense of a canonical afﬁne structure on the smooth circle obtained by glueing the ends of a fundamental domain of the iterate n+1 f .However, if f is cylinder renormalizable with period q , the afﬁne structure on the quotient circle comes from the quotient C /f C/Z.The key here is a fundamental fact of complex analysis, known under a slightly different guise as Liouville’s Theorem: an analytic manifold structure on C/Z is necessarily afﬁne. The second advantage of R over R is closely related to the ﬁrst one: as we cyl have seen, it extends to analytic maps which are not symmetric with respect to the circle, thereby becoming an analytic operator on a complex Banach manifold. In the following section we will show that Epstein critical circle maps are always cylinder renormalizable. 7.2. Cylinders of commuting pairs in the Epstein class In this section we will relate the cylinder renormalization deﬁned above with the renormalization of critical commuting pairs in the Epstein class. The crucial observation is the following: Lemma 7.6. —Let ζ = (η, ξ) be a commuting pair in the Epstein class with χ(ζ) =∞.Denote D the domain of the branched triple covering map η : D → C . η η η(I ) Then the following holds: + − ± • there exist two compex conjugate ﬁxed points p , p of the map η with p ∈±H, η η η and an R-symmetric simple arc l connecting these points such that l ⊂ D and + − η(l) ∩ l ={ p , p }; η η 28 MICHAEL YAMPOLSKY • denoting C the R-symmetric domain bounded by l and η(l) we have C ⊂ D , ζ ζ η + − and the quotient of C ∪ η(C ) \{ p , p } by the action of ζ is a Riemann surface ζ ζ η η homeomorphic to the cylinder C/Z (that is, C is a fundamental crescent of η); • the dependence ζ → C is continuous with respect to the Caratheod ´ ory topology on the Epstein class and the Hausdorff topology on the image. Proof. — Let us ﬁx s>0 such that ζ ∈ E . In view of Lemma 3.2, there exists N = N(s) such that every pair ζ ∈ E with χ(ζ) >N satisﬁes the con- ditions of the existence of Douady coordinates (Lemma 6.2). We may choose as C a fundamental domain for η| , by Theorem 6.4 the choice can be made ζ U continuously. −1 Letus addressthe case χ(ζ) ≤ N(s).Let η : H → H denote the inverse −1 branch which preserves the real axis, η (H) H. By the Denjoy-Wolf Theorem this branch has a unique ﬁxed point p ∈ H ∪ η(I ) which is globally attracting + −1 in H. In this case necessarily Im p >0.Extend η to −H by symmetry. This − + branch has a complex-conjugate ﬁxed point p = p . In view of their unique- η η ness, these points depend continuously on the map η. Note that for a converg- ing sequence of commuting pairs with χ(η) →∞ these ﬁxed points converge to a parabolic ﬁxed point on the real axis. Suppose that the commuting pair ζ does not possess a fundamental cres- cent C . As noted in the proof of Proposition 7.3, in a neighborhood of p the map η becomes a translation in log-linearizing coordinate. Therefore if l is a curve ± ± connecting the ﬁxed points p and the image η(l) \{ p } is disjoint from l,then η η l can be modiﬁed near the ends in such a way that the quotient of the cres- cent bounded by l, η(l) has the conformal type of a bi-inﬁnite cylinder. Thus, for every choice of an R-symmetric curve l connecting the ﬁxed points p the image η(l) \{ p } intersects with l.To ﬁx the ideas, let ξ(0) >0. Consider the maps η ∈ E given by η = η + α for α >0.Let α be the smallest value for which α α 0 the map η has a ﬁxed point, which is then necessarily parabolic. For all α < α α 0 sufﬁciently close to α the map η has a fundamental crescent by Lemma 6.2. 0 α Therefore there exists a value α between 0 and α such that η = η has the ∗ 0 ∗ α following property. For any simple R-symmetric arc l connecting the ﬁxed points −1 of η the intersection (η (l) \{ p }) ∩ l is non-empty, and there is a curve l which ∗ ζ −1 touches η (l) without crossing it. −1 ± ˆ ˆ ˆ Consider such a curve l and denote γ = l ∩ η (l) \{ p }. Again appealing ∗ η to the local picture of the dynamics near the ﬁxed points and near the real axis we seethat wemay choose l for which the set γ is positive distance away from p and R.Let us now ﬁx l for which γ has the smallest Euclidean diameter. Enclose γ into a Euclidean disk E with a diameter equal to diam γ,and take apoint x ∈ γ ∩ ∂E.Let x , x , ... denote the consecutive η -preimages of x lying −1 −2 ∗ HYPERBOLICITY OF RENORMALIZATION 29 in γ . Note that this sequence has to be ﬁnite, since otherwise the set ∪x ⊂ γ is −i −1 compact and invariant under η and thus contains a ﬁxed point, which contradicts ± −1 the fact that p are the only ﬁxed points of η .Let x be the last term in this −n η ∗ −1 sequence. Since η (x )/ ∈ γ , we may slightly modify the curve l near x ,so −n −n that the new curve and its preimage become disjoint in a neighborhood of x . −n ˜ ˆ Proceeding in this fashion, we obtain a curve l which is a small perturbation of l −1 ˜ ˜ such that l ∩ η (l) ⊂ E \ D (x) for some >0. Repeating the procedure a ﬁnite number of times we arrive at a curve l with −1 ± ˇ ˇ diam l ∩ η (l) \ p < diam γ, ∗ η contradictory to the way in which l was selected. Tracing back the assumptions, we see that for any ζ ∈ E with χ(ζ) =∞ the arc l may be chosen with η(l) ∩ l ={ p }. The continuity of the construction is ensured in the same way as in Proposition 7.3. Given ζ = (η, ξ) ∈ E and C as above, denote R (z) the ﬁrst return map ζ C of the crescent C .Let c ∈ T be the only critical point of R and denote ζ ζ C π : C ∪ η(C ) → C/Z the projection speciﬁed by c → 0.Inanobvious fashion C ζ ζ ζ we have: Proposition 7.7. — The projection of the ﬁrst return map −1 f = π ◦ R ◦ π C C C ζ ζ ζ is deﬁned and analytic on an open neighborhood of the circle T,and f | is an analytic C T critical circle map. Let us show that this critical circle map does not, in fact, depend on the particular choice of the fundamental crescent C : Proposition 7.8. —Let C ⊂ D be a different domain satisfying the conditions of ζ ζ Lemma 7.6. Then f | ≡ f | . C T C T Proof. — To ﬁx our ideas, assume that ξ(0) >0.Denote O ⊂ C the smallest full set in the plane containing C ∪ C .The set O is bounded by two curves l , ζ 1 −1 l meeting at p ,let l ∩ R <l ∩ R. By construction, the curves l and η (l ) 2 1 2 1 1 bound a fundamental crescent C . The Denjoy-Wolff Theorem together with the dynamics of η| implies that for every point z ∈ C there is the smallest n(z) ≥ 0 −n(z) −n(z) such that η (z) ∈ C .Deﬁne h(z) ≡ η (z).The value of n(z) is constant on ζ 30 MICHAEL YAMPOLSKY n(z) an open neighborhood of z except for z ∈ η (∂C ) for which the values on the −1 two sides of ∂C differ by 1. Therefore the projection ψ = π ◦ h ◦ π is an ζ C −1 analytic endomorphism of C/Z.Since η is univalent, ψ has no critical points, and since η is monotone, the degree of ψ| is one. Hence, ψ is a conformal R T map. Also, ψ(0) = 0 by construction of the projections π and π . Therefore, C C −1 ψ ≡ Id. By deﬁnition of the ﬁrst return map, ψ ◦ f ◦ψ = f on a neighborhood C C of T,so f = f in that neighborhood. The equality f = f is shown in the C C C C ζ ζ ζ same way, and the claim follows. In view of the last lemma, on a neighborhood of T we obtain a well-deﬁned analytic critical circle map f ≡ f , ζ C which we will refer to as the cylinder map of ζ . In Figure 4 we illustrate the action of the map f near T. Schematically shown is the preimage of the circle T ⊂ C/Z which consist of the circle itself, together with analytic arcs emanating from 0 at angles kπ/3 to T. Analytic continuations of these preimages may eventually branch further, as shown. FIG. 4. 7.3. The cylinder renormalization operator Proposition 7.9. —For every s ∈ (0, 1) there exists an equatorial neighborhood U = U(s) such that for every commuting pair ζ ∈ E with χ(ζ) =∞ the cylinder map f is s ζ in C . U HYPERBOLICITY OF RENORMALIZATION 31 Proof. — First, denote mod ζ = sup mod U over all equatorial annuli U such that f ∈ C . To prove the claim, let us assume the contrary. Then there is ζ U a sequence ζ ∈ E with χ(ζ ) =∞ such that mod(ζ ) → 0. Using compactness n s n n of E as stated in Lemma 3.2 we may pass to a subsequence ζ → ζ ∈ E .The s n s height of the commuting pair ζ is necessarily inﬁnite. The dynamics of ζ produces an analytic return map h from an equatorial neighborhood in the repelling Fatou cylinder of η to the attracting Fatou cylinder. Again by Lemma 3.2, there exists δ = δ(s) >0 such that h ∈ C with modU> δ. By Theorem 6.4 for any ζ ζ U n sufﬁciently close to ζ , mod ζ > δ, and we have arrived to a contradiction. For the remainder of this paper let us ﬁx the value of s to be the universal constant from Lemma 3.4. Let us say that two pairs ζ , ζ in E are conformally 1 2 s equivalent, ζ ∼ ζ ,if χ(ζ ) = χ(ζ ) =∞ and f | ≡ f | . The name of the equiv- 1 conf 2 1 2 ζ T ζ T 1 2 alence relation is explained by the following: Proposition 7.10. — Suppose ζ ∼ ζ . Then there exists a conformal map ψ from 1 conf 2 a neighborhood of I to a neighborhood of I which conjugates the two Epstein commuting ζ ζ 1 2 pairs. Conversely, assume that there is a conjugacy ψ, whose domain includes a fundamental crescent C .Then ζ ∼ ζ . ζ 1 conf 2 Proof. — The ﬁrst claim is quite obvious, since the identity map of C/Z lifts via the projections π , π to a conformal conjugacy between the two commut- C C ζ ζ 1 2 ing pairs. On the other hand, any such conjugacy ψ pojects to a conformal map h between equatorial annuli, which conjugates f , f .Ifthe domain of ψ contains ζ ζ 1 2 C ,then h extends to a conformal endomorphism of C/Z.Since h(0) = 0,in this case h ≡ Id. Recall that S ={ζ ∈ C| χ(ζ) = r}. The equivalence relation ∼ is clearly r conf closed. By Lemma 7.6 we have: Proposition 7.11. —For every r ∈ N the correspondence ι : (S ∩ E )/ ∼ → C r s conf given by [ζ ] → f is continuous. ∼ ζ conf Moreover, Proposition 7.12. —For every s ∈ (0, 1) and r ∈ N there exists c = c(s, r) >0 such 0 R that the following holds. If we equip the Epstein class E with C -distance, and C with the sup-distance, then for every ζ ,ζ ∈ E ∩S we have dist 0 (ζ ,ζ ) ≥ c(s, r) dist 0 ( f , f ). 1 2 s r C 1 2 C ζ ζ 1 2 Proof. — For ζ = (η, ξ) ∈ E let φ denote the uniformizing coordinate C → C/Z,and Φ : C → C be its lift. By Lemma 7.6 the crescent C moves ζ ζ ζ continuously in E . On the other hand, by Lemma 3.2, E is Caratheodory com- −1 pact, and hence the domain C ∪ η(C ) ∪ η (C ) has a bounded shape near ζ ζ ζ 32 MICHAEL YAMPOLSKY −1 C ∩ R.Since Φ conformally extends to C ∪ η(C ) ∪ η (C ), we may apply the ζ ζ ζ ζ Koebe Distortion Theorem to it to conclude that the distortion of Φ on C ∩ R is bounded by a constant which depends only on s and r.Moreover, by real apriori bounds the derivative Φ is bounded by const(s, r).Noting that f | is the ζ T projection of the composition η ◦ ξ , we have the desired estimate. Proposition 7.13. —Let U be as in Proposition 7.9. There exists N = N(U) ∈ N such that the following holds: (I) suppose ζ ∈ E is at least N-times renormalizable. Then the renormalization R ζ ∈ E ; (II) also R ζ has a holomorphic pair extension H with universal bounds as in Theo- rem 5.4; (III) denote H the holomorphic pair extension of pR ζ which is the linear rescaling −l of H . Then there exists an iterate η which conformally conjugates the pair H to a holomorphic pair G with ∆ C for some fundamental crescent C ; G ζ ζ (IV) moreover, π : ∆ → C/Z ∩ U; C G (V) ﬁnally, there is a fundamental crescent C N−1 compactly contained in U (cf. pR f Figure 4). Proof. — By real apriori bounds, eventually all renormalizations of a com- muting pair ζ ∈ E lie in E . The existense of N as in (I) follows from Lemma 3.2. Statement (II) is a consequence of Theorem 5.4 and Lemma 3.2. Claims (III) and (IV) follow since the range of the holomorphic pair H is commensurable with I by Theorem 5.4. Finally, (V) follows from Lemma 7.6 and Lemma 3.2. Deﬁnition 7.2. — For the remainder of the paper ﬁx U as in Proposition 7.9 and set M = C , N = N(U). Condition (V) of Proposition 7.13 implies that f is a renormalizable critical cylinder map. Let f be its renormalization corresponding to the crescent C N−1 .We pR f will call it the cylinder renormalization of f , and write f ≡ R f . cyl ζ It follows from the ﬁrst condition of Proposition 7.13 that R f ∈ C . Therefore, by cyl ζ U Lemma 7.3, for every pair ζ as above, the tranformation f → R f extends to an open ζ cyl ζ neighborhood Y ⊂ C as an analytic operator Y → C . We shall call this operator the U U cylinder renormalization operator. The deﬁnition of R implies the following connection with the usual renor- cyl malization operator: Proposition 7.14. — In the above notation, we have R f ≡ f N . cyl ζ R ζ HYPERBOLICITY OF RENORMALIZATION 33 tq Proof. — By condition (V) of Proposition 7.13, there is an iterate f ,for −1 some t ∈ Z, which conformally maps π (C N−1 ) to C N .Thisconformal map R f R ζ C ζ projects to a conjugacy ψ : C/Z → C/Z between R f and f N .Since ψ is cyl ζ R ζ conformal, and ψ(0) = 0, necessarily ψ ≡ Id. Moreover, Proposition 7.15. —If ζ , ζ are at least N-times renormalizable elements of E ,and 1 2 s N N ζ ∼ ζ ,then R ζ ∼ R ζ . Hence the action of R is well-deﬁned on the quotient space 1 conf 2 1 conf 2 E / ∼,and s conf ι ◦ R = R ◦ ι. cyl Proof. — The claim follows from condition (V) of Proposition 7.13, similarly to the preceding Proposition. 8. Local stable manifold of a periodic point of R cyl Let a commuting pair ζ be a periodic point of the renormalization opera- tor R. As follows from Lemma 3.4, ζ ∈ E . Naturally, ζ is also a periodic point N Nm of R , and we may ﬁx the smallest m ∈ N such that R (ζ) = ζ . As seen from Proposition 7.15, the critical circle map f = f ∈ M is a periodic point of the cylinder renormalization operator: ˆ ˆ R f = f . cyl ˆ ˆ Letusﬁxa periodic point f .Set ρ = ρ( f ),and deﬁne D ={ f ∈ M,such that ρ( f ) = ρ}. Again from Proposition 7.15 we have Proposition 8.1. — There exists a neighborhood Y of f in M such that for every im im f ∈ Y ∩ D , R f is deﬁned for all i,and R f → f uniformly in Y. cyl cyl Proof. — Assume that the statement of the proposition does not hold. Since R is deﬁned in an open neighborhood of f , the assumption implies that there cyl exists δ >0 such that for every i ∈ N and every >0 there exists f such that i, im ˆ ˆ dist( f , f ) < ,and dist( f , R f ) > δ. Select a sequence → 0 and denote i, i, i cyl f = f .Let λ be the length of the domain of (imN − 1)-st pre-renormalization i i, i imN−1 of f : λ =|I |, and consider the inﬁnite sequence i i pR f −1 imN−1+j T = λ pR f (λ z) . i i i j=−imN−1 By complex apriori bounds (Theorem 5.4) for each i there exists such N that for all n ≥ N the renormalization R f extends to a holomorphic commuting pair in i 34 MICHAEL YAMPOLSKY H(µ) with a universal bound µ.By Remark 5.1,the valueof N can be chosen uniformly for all i. By Lemma 5.3 we may select a subsequence {i } such that the sequences T converge to a limiting renormalization tower T with complex apriori bounds. Denote T the limiting renormalization tower corresponding to the periodic point ζ . By Proposition 7.12, the distance between the base pairs of T and T is at least const ·δ. This contradicts the Tower Rigidity Theorem. Below we shall demonstrate that the local stable set of f is a submanifold of M: Theorem 8.2. — There is an open neighborhood W ⊂ M of f such that D ∩ W is a smooth submanifold of M of codimension 1. Fix an equatorial neighborhood U U such that f analytically extends to U . As before denote p /q the reduced form of the k-th continued fraction ˆ k k convergent of ρ.Furthermore,deﬁne D as the set of g ∈ M for which ρ(g) = p /q k k k and 0 is a periodic point with period q . As follows from the Implicit Function Theorem, this is a codimension 1 submanifold of M. As seen in the previous section, the tangent space of a point in C may be identiﬁed with B ,and that U U R R of a point in M with the real slice B . Let us deﬁne a cone C ∈ B as U U C = v ∈ B , such that inf v(x) >0 . x∈R Lemma 8.3. —Let f ∈ D , and denote T D ⊂ B the tangent space at this point. k f k Then T D ∩ C =∅. f k Proof. — Let v ∈ C and suppose { f }⊂ M is a one-parameter family such that −1 −1 π ◦ f ◦ π = π ◦ f ◦ π + tv + o(t). −1 −1 k q Then for sufﬁciently small values of t, π ◦ f ◦π > π ◦ f ◦π.Hence f (0) = f (0) and thus f ∈ / D . t k Now let f be as above. Elementary considerations of the Intermediate Value Theorem imply that for every k there exists a value of θ ∈ (0, 1) such that the map f = R ◦ f ∈ D .Moreover, if we denote θ the angle with the smallest θ θ k k absolute value satisfying this property, then θ → 0.Hence, for all k large enough R (U ) ⊃ U,so f ∈ M.Set f = f and let T = T D ⊂ B .Fix v ∈ C.By θ ˆ θ k θ k f k k f k k k U Lemma 8.3 and the Hahn-Banach Theorem there exists >0 such that for every R ∗ k there exists a linear functional h ∈ (B ) with ||h || = 1,such that Ker h = T k k k k U HYPERBOLICITY OF RENORMALIZATION 35 and h (v) > . By the Alaoglu Theorem, we may select a subsequence h weakly k n R ∗ converging to h ∈ (B ) . Necessarily v ∈ / Ker h,so h ≡ 0.Set T = Ker h. Proof of Theorem 8.2. — By the above, we may select a splitting B = T⊕v·R. R R Denote p : B → T the corresponding projection, and let ψ : M → B be U U a local chart at f . Lemma 8.3 together with the Mean Value Theorem imply that p ◦ ψ : D → T is an isomorphism onto the image, and there exists an open neighborhood U of the origin in T,suchthat p ◦ ψ(D ) ⊃ U.Since thegraphs D are analytic, we may select a C -converging subsequence D whose limit is k k a smooth graph G over U . Necessarily, for every g ∈ G, ρ(g) = ρ.Aswehave seen above, every point g ∈ D in a sufﬁciently small neighborhood of f is in G, and thus G is an open neighborhood in D . 9. Proof of the main result As in the previous section, let f : U → C/Z,where U U,beaperiodic ˆ ˆ f f point of R with period m.Set ρ = ρ( f ) and let W be as in Theorem 8.2. cyl Denote T ≡ T (D ∩ W) the tangent space to the codimension one submanifold ˆ ρ ˆ ˆ at f ,and let L be the differential of R at f : cyl m R R L = D R : B → B , L : T → T f cyl U U Recall that a continuous linear operator on a Banach space is called compact if it maps the closed unit ball of the space onto a compact set. This condition is equivalent to the image of every closed bounded set being compact. m R R Proposition 9.1. — The operator L = D R : B → B is compact. f cyl U U Proof. — Denote B the unit ball in B and let v ∈ B . By deﬁnition of 1 1 R , the image L v is an analytic vector ﬁeld on U U.Let C U be cyl ˆ ˆ f f the fundamental crescent corresponding to R f ,and φ = π : C → C/Z the cyl ˆ ˆ f f −1 uniformizing coordinate. The ﬁrst return map R of φ (U) ⊂ C consists of a ﬁnite collection of iterates f . By compactness considerations, the vector ﬁeld v which is the perturbation of each of these iterates induced by v has norm bounded by −1 a constant C independent of v.Since φ is also bounded in φ (U), there exists D>0 independent of v such that ||L v| || <D. Since the analytic functions in U whose norms in U are bounded by D form a normal family in U, the image L (B ) is compact. Hyperbolicity: stable direction. — By Proposition 8.1, sup ||L | || < ∞. k 36 MICHAEL YAMPOLSKY Let us show that for some iterate the norm is strictly less than one: Proposition 9.2. — There exists k ∈ N such that ||L | || <1. Proof. — Assume the contrary. Then the operator L | has a unit spec- tral radius. The discreteness of the spectrum of a compact operator implies that Sp(L | ) ∩ B =∅, and moreover, since every non-zero element of the spectrum T 1 of a compact operator is an eigenvalue, there is v ∈ T such that L (v) = v.The s c s space T may then be split into a direct sum E ⊕ E such that E , the stable space, is a closed L -invariant subspace such that ||L | s || <1 for some ∈ N, and E , the central space, is also L -invariant, ﬁnite dimensional, and for every c c k v ∈ E , inf ||L v|| >0.The space E is spanned by solutions of (T − λ Id) v = 0, ∈N k ∈ N, λ ∈ B . If we choose a small perturbation f ∈ D ∩ W of f in the direction 1 ρ c m of E close enough to f , then the dynamics of R on f is going to be dominated cyl km by the central part L | . In particular, the rate of convergence R f −→ f can cyl k→∞ not be geometric, contrary to Theorem 4.3. Hyperbolicity: unstable direction. — As before, let B , such that inf v(x) >0 . C = v ∈ x∈R The key properties of C are listed in the next lemma: Lemma 9.3. (I) The cone C is renormalization-invariant: L : C → C , (II) Moreover, there exists α >0 and k ∈ N such that for any vector ﬁeld v ∈ C inf L (v(x)) > (1 + α) inf v(x), x∈R x∈R (III) Finally, if v ∈ B belongs to the closure of C,and v = 0, then there exists ∈ N for which L (v) ∈ C . −1 ˜ ˆ Proof. — Let f = π ◦ f ◦ π . As follows from an elementary computation, if ˜ ˜ f (x) = f (x) + tv(x) + o(t),then 2 2 ˜ ˜ f (x) = f (x) + tv (x) + o(t) (9.1) ˜ ˜ ˜ ˜ = f (x) + t( f ( f (x))v(x) + v( f (x))) + o(t), and more generally, if we write n n ˜ ˜ f (x) = f (x) + tv (x) + o(t), (9.2) n−1 n−1 then v (x) = f ( f (x))v (x) + v( f (x)). n n−1 HYPERBOLICITY OF RENORMALIZATION 37 Let us denote C U the fundamental crescent of f corresponding to the k-th cylinder renormalization R f ,and J = C ∩ T.Let π (x) be the corresponding k k k cyl uniformizing coordinate C → C/Z,and Φ : C → C its lift. In the local chart k k n −1 given by π ,the k-th cylinder renormalization R f is represented by the iterate k cyl mk g = f for some m ∈ N. To prove the ﬁrst claim, observe that by (9.2), (9.3) inf v (x) ≥ inf v(x) >0. x∈R x∈R The image −1 (9.4) L v =[(Φ ) ◦ g · (v | )]◦ (Φ ) . k k q J k mk k Since (Φ ) >0, (I) follows. To prove(II), notethat by the real apriori bounds, the sizes of the intervals J decrease geometrically with k. On the other hand, applying the Koebe Distortion Theorem to the conformal extension of Φ to C and its two neighboring iterates, k k we see that the distortion of Φ on the interval g ( J ) is bounded uniformly in k. k k k Hence, there exists α >0 and l ∈ N such that for all k ≥ l (Φ ) | >1 + α, k g ( J ) k k and (II) follows from (9.4). Finally, to see (III), observe that if v(x) ≡ 0, then there exist , n ≤ q such that v(x) >0 for every x ∈ f ( J ). The claim now follows from (9.2). We conclude: Proposition 9.4. — There exists an eigenvector v ∈ C : L (v ˆ) = δv ˆ, where |δ| >1. Proof. — Part (II) of the previousLemma impliesthatthe spectral radius R (L ) >1. Since the spectrum of a compact operator is discrete, and its every Sp non-zero element is an eigenvalue, the claim follows. Universality: Proof of Corollary 4.5. — The second claim is an obvious corollary of the main theorem, the proof of the ﬁrst one follows. Denote ζ ∈ E the ﬁxed point of ∗ s R with ρ(ζ ) = ( 5 − 1)/2,and set f = ι(ζ ), deﬁned in some annulus U ⊃ U.By ∗ ∗ ∗ f Theorem 4.4, there exists m ∈ N such that f is a hyperbolic ﬁxed point of R with ∗ cyl loc one-dimensional unstable direction. Let W be its local stable manifold, set M = mN, and denote δ >1 the modulus of the unstable eigenvalue. Set P ={ f ∈ C , such that ρ( f ) = [1, ..., 1], and f has a periodic orbit with eigenvalue ± 1}. 38 MICHAEL YAMPOLSKY By the Implicit Function Theorem, P is a local submanifold of C of codi- mension one. Moreover, in a neighborhood of f , R : P → P .Let C be ∗ cyl n n−M a subcone of C with C ∩ B compact. Then by Theorem 4.4, there exists an open neighborhood W ⊂ C of f and K ∈ N,such that for k> K, P ∩ W is U ∗ loc agraph over W transversal to C at every point. Moreover, for every n> K the ∞ loc M sequence {P } converges to W at a geometric rate δ . For our purposes n+lM l=1 s we will set C = D R (C ). ∗ cyl Let V ⊃ T be an equatorial annulus chosen so that f ∈ C .By compex µ V apriori bounds, there exists N ∈ N such that for all n> N the renormaliza- 1 1 tion R f extends to a holomorphic commuting pair with universal bounds. Since R f → ζ ∈ E and by Lemma 7.6, there exists N ≥ N such that for all k> N µ ∗ s 2 1 2 the map f ∈ C is cylinder-renormalizable with period q , and the corresponding µ V k cylinder renormalization g is in C . Fix some such k. By Proposition 7.3, there k U is a neighborhood G of f in C such that the cylinder renormalization f → g µ V µ k ∗ ∗ extends to an analytic operator Z : G → C .Provided G is sufﬁciently small, there t t is an iterate R (G) ⊂ W.Denote {h }⊂ C the smooth family R (Z({ f }∩ G)). µ U µ cyl cyl R R Let u ∈ B be the tangent vector to the family f at f ,and v ∈ B its image µ µ V ∗ U D (R ◦ Z)u. As in Lemma 9.3 we have inf D Zu > 0 and hence v ∈ C . f f µ cyl µ ∗ ∗ Note that the curve {h for µ ∈ I } for i large is bounded by a pair of points µ i+lM ± ± f ∈ P ,where n = i + lM − k − tM, and the claim follows from the transversality n n property of P . 10. Concluding remarks Let us conclude by discussing some remaining open problems in the ﬁeld. Of primary importance to us is bringing Lanford’s Program to a completion. Passing from the hyperbolicity of periodic orbits of R to the hyperbolicity of the whole cyl renormalization horseshoe requires, as a preliminary step, endowing the local sta- ble sets of the points in the horseshoe with the structure of analytic submani- folds. In addition, the case of unbounded type necessitates analyzing the situation when the periods of renormalization grow without a bound, and showing that the rate of convergence to the horseshoe remains uniformly bounded in this situation. We recently carried out the relevant analysis in a joint paper with A. Epstein [EY]. We extend the result of this paper to the whole horseshoe in a forthcoming work [Ya3]. Commonly, renormalization convergence results in one-dimensional dynamics are connected with rigidity properties of the renormalized maps. The problem 1+α relevant to our study is a C rigidity question: When is the conjugacy between two critical circle maps with the same irrational rotation 1+α number, which maps 0 to 0, C -smooth? HYPERBOLICITY OF RENORMALIZATION 39 The study of this question is the main subject of the papers of de Faria and de Melo [dFdM1,dFdM2]. These authors demonstrate that the conjugacy has the desired smoothness in the case when both maps are analytic, and the type of 1+α the rotation number is bounded. They prove that, with these restrictions, the C rigidity is equivalent to the geometric rate of convergence of renormalizations, and derive their result from their Theorem 4.3. On the other hand, they show that if both of the conditions are relaxed, the statement does not hold. As we described above, we expect our methods to bring Lanford’s Program to a completion, and in particular, to extend Theorem 4.3 to unbounded types. It is then an intriguing problem, whether the answer to the above question is positive when the two maps are analytic, but the rotation numbers are unbounded. The situation when the type is bounded, but the maps are only smooth also requires study. In this paper we have introduced a new renormalization for critical circle maps. We hope that the reader has been convinced that apart from enabling us to solve the hyperbolicity problem, this new construction is more natural. It would be instructive therefore to re-do the proof of Theorem 4.1 using R instead of R. cyl This mainly requires understanding the geometry of a periodic point of R to cyl deﬁne an appropriate analogue of the Epstein class, and ﬁnding the corresponding replacements of de Faria’s holomorphic commuting pairs. For the moment, the only situation when we are able to carry out the whole theory without recourse to commuting pairs is the degenerate case of parabolic renormalization, studied in [EY]. Let us also comment that our method should be applicable to the study of universality of Siegel disks (see e.g. [MP]) in the situation when complex apriori bounds exist (such as Siegel quadratics of bounded type, cf. [Ya1]). This problem is analogous to the study of renormalization of non-real quadratic polynomials and the related aspects of the geometry of the Mandelbrot set, and should lead to the construction of hyperbolic non-real periodic orbits of R . cyl REFERENCES [BR] L. BERS and H.L.ROYDEN, Holomorphic families of injections, Acta Math., 157 (1986), 259–286. [Do] A. DOUADY, Does a Julia set depend continuously on the polynomial?, in Complex dynamical systems: The math- ematics behind the Mandelbrot set and Julia sets, R. L. Devaney (ed.), Proc. of Symposia in Applied Math., Vol. 49, Amer. Math. Soc., 1994, pp. 91–138. [DH1] A. DOUADY and J. H. HUBBARD, Etude dynamique des polynˆ omes complexes, I-II, Pub. Math. d’Orsay, 1984. [DH2] A. DOUADY and J. H. HUBBARD, On the dynamics of polynomial-like mappings, Ann. Sci. Ec. Norm. Sup., 18 (1985), 287–343. [dF1] E. DE FARIA, Proof of universality for critical circle mappings, Thesis, CUNY, 1992. [dF2] E. DE FARIA, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergodic Theory Dynam. Systems, 19 (1999), no. 4, 995–1035. [dFdM1] E. DE FARIA and W. DE MELO, Rigidity of critical circle mappings I, J. Eur. Math. Soc. (JEMS), 1 (1999), no. 4, 339–392. 40 MICHAEL YAMPOLSKY [dFdM2] E. 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Th. & Dyn. Systems, 19 (1999), 227–257. HYPERBOLICITY OF RENORMALIZATION 41 [Ya2] M. YAMPOLSKY, The attractor of renormalization and rigidity of towers of critical circle maps, Commun. Math. Phys., 218 (2001), no. 3, 537–568. [Ya3] M. YAMPOLSKY, The global horseshoe for the renormalization of critical circle maps, Preprint, 2002. [Yoc] J.-C. YOCCOZ, Il n’ya pas de contre-example de Denjoy analytique, C.R. Acad. Sci. Paris, 298 (1984) s´ erie I, 141–144. M. Y. Department of Mathematics University of Toronto Toronto, Ontario M5S 3G3 Canada yampol@math.utoronto.ca Manuscrit reçu le 5 décembre 2000.

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Hyperbolicity of renormalization of critical circle maps

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Hyperbolicity of renormalization of critical circle maps

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