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Hénon mappings in the complex domain I: The global topology of dynamical space

Hénon mappings in the complex domain I: The global topology of dynamical space HI'NON MAPPINGS IN THE COMPLEX DOMAIN I: THE GLOBAL TOPOLOGY OF DYNAMICAL SPACE by JOHN H. HUBBARD and RAI,VI-I W. OBERSTE-VORTH CONTENTS 1. Introduction ............................................................................. 5 2. An algebraic characterization of H6non mappings ............................................. 9 3. Solenoidal mappings ...................................................................... 11 4. Embeddings of the solenoid in S 8 .......................................................... 18 5. The functions G~ and $& ................................................................ 24 6. The global topology of H~non mappings ................................................... 28 7. The foliations of U+ ..................................................................... 32 8. An analytic description of U+ ............................................................. 35 9. The canonical compactification of K+ ...................................................... 40 I~FE~NCES ................................................................................. 45 1. Introduction In 1969, H6non ([H6I] and [H62]) began the investigation of the mappings F: ~ , where a+0, as mappings having roughly the same behavior as a particular Poincar~ section of the Lorenz differential equation. H~non demonstrated numerically that for certain values of the parameters the mappings appeared to have a strange attractor. This has finally been established rigorously by Benedicks and Carleson ([BC], [MV]). There has since been an enormous amount of work on the dynamics of the H~non mappings (in particular, see [Ho], [HWh] and [HWi], which give further references). This work is all in the real domain. As far as we know, this paper ([H] was an early version) is the first attempt to understand the H~non mappings in C ~'. Recently others have done work in this area including Friedland and Milnor ([FM] and [M1]), Bedford, Lyubich, and Smillie ([B], [BS1], [BS2], [BS3], [BS4], [BLS], IS]), and Formess and Sibony ([FS]). In the study of iteration of polynomials of one variable, extending to complex values of the variable has been very useful, even when the original polynomials were JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH real. We hope that the same thing will happen here, more or less for the same reason. There is essentially nothing that can be said about real polynomials which is independent of the coefficients, largely because virtually all features independent of conjugation, such as periodic cycles, are likely to disappear under perturbation. In the complex domain, the behavior is far more uniform. Our work started from a different point of view. In 1982, Calabi suggested that the computer should be used to investigate the basin of attraction of one of the two attractive fixed points of the mapping (11 ) (;) The reason for examining this was that it provided an example of a Fatou-Bieber- bach domain. These are open subsets U C C" which are biholomorphically isomorphic to C" and whose complement, C ~ -- U, has non-empty interior. When an automorphism of C" has an attractive fixed point or attractive cycle, the basin is always such a domain. Fatou and Bieberbach ([17], [Bi]) first constructed examples of such domains as basins of attractive fixed points. They have been extensively studied in [BS2] and [FS]. 9 Despite considerable numerical work, we were unable to work out the topology of the closures of the basins and decided to look at simpler automorphisms of C ~, with quadratic polynomials as coordinates. Section 2 shows that the Htnon family encompasses a significant part of this family. The east of players. Most of the work on Htnon mappings in the real case has focused on attractors. In the complex, attractors are uninteresting since the only attrac- tors are points. The invariant subsets considered here are inspired by the dynamics of polynomials, as explained below. For any mapping f, letf ~ denote the n-fold composition off or f-1 depending on whether n is positive or negative. Our approach has been inspired by the study of complex polynomials of a single variable. Given a polynomial p(z), the natural set to study is K~ = { z [p~ does not tend to oo as n ~ ~ } and its boundary J~ = 0Kr, also known as the Julia set ofp. Another definition of J~ is J~ = { z [ on no neighborhood of z is the sequence { pO, } normal }. The sets studied here are defined in imitation of the one-dimensional case. For a Htnon mapping, the obvious generalization of the Julia set is J~ ={(;)~ n~ neighb~176176 ~ (;) is the sequence { F~  } n~ } ' HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I where a sequence of functions on U C (I 3 with values in C ~' is defined to be normal if every subsequence has a subsequence which converges uniformly on compact subsets to a function with values in p2, the complex projective plane. Define for a Hdnon mapping F the following sets: = C 2 -- K+ [l o (;Ill oo-o,,en o } and -~C2--K . Further, define J = OK:~, K ----- K+ c~ K_, and J = J+ n J_. It will be seen that K and J are compact and of course invariant under F. These are the spaces which we most wish to understand. Ma~n results. This paper contains three main results: two concern the structure of C 2 -- K~. Topologically, we will show that this set is homeomorphic to a fibration over the reals with fiber a 3-sphere with a solenoid removed (Theorem 6.1). Analytically, C 2 -- K is isomorphic to a quotient of (C -- D) X C, where D C C is the unit disc, by a group of automorphisms which we determine explicitly (Section 8). The third result gives a compactification (I 2 to which the Hrnon mappings extend canonically, analogous to compactifying C by adding a circle at infinity (Theorem 9.1). The proofs of these results require both some analydc and some topological pre- liminaries. Most of the topology (Sections 3 and 4) concerns solenoidal mappings, one of which plays much the same role with respect to Hrnon mappings as multiplying angles by d does for iteration of polynomials. We go into more details than is strictly necessary for our purposes, but we feel that viewing the surrounding countryside makes our pard- cular mappings easier to understand, and the classification of solenoidal mappings (Theorem 3.10) is of independent interest. For the analytical results, the most important construction is the analog of the Brttcher coordinate ([M2]). When p is a monic polynomial, this is the function ?~ defined in a neighborhood of o0 such that and ~(z) = z + 0(1) near oo. The function q~(z) is constructed by making sense of the following O~(z) = lim (p~ n ..-~ oo This is a standard scattering theory construction: go toward oo via p and return via the unperturbed mapping z ~ z d. The fractional power is not a priori defined, and has to be dealt with carefully. JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH The problem of the branches of the roots can be circumvented by defining G~(z) = lirno ~; log+ Ip~ where log+(x) = sup{log(x), 0 }, which is the Green's function of K~. This construction generalizes for H6non mappings as follows. Let a subscript I or 2 denote the projection onto the first or second coordinate as in (F~ = pr t o F ~ Now define the limits (;) 1 G -= lim ~ log+ II F~  "(x) II, of course, the matter of where these are defined and the convergence of the limits must be dealt with (and are, in Section 5). Since the first version of this paper was written, much further work on G has been done, more particularly by considering the closed (1, 1)-currents ~  = dd ~ G+ which are analogs of the Brohlin measure ([BS1], [FS]). The measure ~ = ~+ ^ ~_ has also turned out to be very important. As far as we know, the complex analytic mappings ? have not received similar attention, but they are even more important to our development. More particularly, the argument of the B6ttcher coordinate has led to the theory of external angles and is fundamental to the combinatorial study of the dynamics of polynomials ([DH], [T]). When the functions q0 are combined with the compactifica- tion in Section 9, more particularly Corollary 9.4, we find that there is an analogous theory of external angles for H6non mappings; perhaps we can hope to use the techniques using external rays, etc., to combinatorially describe H6non mappings. A case in point is the Benedicks-Carleson result in [BC], where the combinatorics is so reminicent of puzzles and tableaux as in [Y], [BH] and [HY]. Continuations of this paper will present results about Hdnon mappings as pertur- bations of polynomials ([HO]). The paper [O] studies the dynamics of complex horse- shoes using techniques from these papers. Acknowledgments. -- This paper has taken a long time to write and we have benefited from innumerably many conversations. We particularly want to thank A. Douady, J. Smillie, E. Bedford, B. Branner, J. Milnor, D. Sullivan, C. McMullen, J. C. Yoccoz, M. Shishikura, N. Sibony, L. Carleson, M. Benedicks, C. Bardos, T. Bousch, D. Faught, and L. Ma. Much computer experimentation went into the formulation of the theorems, and we thank H. Smith for the immense amount of programming he performed for us over the years. HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I No scientific research is possible without financial support: we thank the National Science Foundation for grant DMS 8901729 and earlier grants, the mathematics depart- ments of Cornell University, Yale University, the University of South Florida, SUNY at Stony Brook, the Universit6 de Paris-Sud, the Institute for Advanced Study, the Geometry Project at the University of Minnesota, the Max Planck Institut i'tir Mathe- matik, the Institut des Hautes Etudes Scientifiques, and the Ecole Normale Supdrieure of Paris and Lyon for support and hospitality during the years when this paper was conceived and written. 2. An algebraic characterization of H~non mappings The family of mappings on C a with quadratic coordinate functions depends a priori on 19 parameters. The H6non mappings = , a4= 0, represent some conjugacy classes of quadratic automorphisms. In this section it is shown that the only other conjugacy classes are represented by the elementary mappings, where k 1 = 1 implies b = 1 and k 2 = 1 implies either b = d+ 1 or d---- 1. Note that the elementary mappings consist of several one- and two-parameter families: A +&t' b,0, b, 1, D = Note that every polynomial mapping, G, of degree 2 can be written in the form 13 =- G0 + G1 + G~, with each G k homogeneous of degree k and that every polynomial automorphism has constant Jacobian determinant. The following theorem gives the Jacob!an Conjecture in this context, i.e., any polynomial mapping of degree 2 with non-zero constant Jacobian determinant is an automorphism. Remark. -- If G~ satisfies the non-degeneracy condition G~-I(0) ----- 0, then the mapping G extends to give an endomorphism of P~, which will be of degree 4. More JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH generally, if the mapping were given by polynomials of any degree d, and the leading terms G a were non-degenerate, then the mapping defines an endomorphism of pa of degree d a. Of course, this is incompatible with G being an automorphism. Since G a is degenerate, G~-l(0) is a line t0, and Ga(C ~) is a line r Either go and gi coincide, in which case the mapping is elementary, or they do not coincide, and the mapping is a genera- lized Hdnon mapping. Theorem 2.1. -- For every polynomial mapping G : C a -+ C 2 of degree ~ with constant non-zero Jacobian determinant, the image of Gz and the set on which Gz vanishes are lines through the origin. If these lines are linearly independent, then G is conjugate to a Hdnon mapping. Otherwise, G is conjugate to an elementary mapping. Proof. -- The general polynomial mapping of degree two is = as+bax+cay+dax 2+eaxy+fayz]" The quadratic terms of the Jacobian determinant generally yield the relations dl ei fi eg fa ' da So the image of the quadratic terms is a line, gl; assume that dz = e9 =fa = O. The linear terms of the Jacobian determinant generally yield the relations e~ ba ca = 4dlf~ be ca. Since G is injective, bz and c a cannot both be 0. So Ga vanishes on a line, t o. Ifg o and ti are linearly independent, then assume di = 1 and el =fi = 0 (sending to to they-axis). The Jacobian condition shows that ca = 0 and this is a Hdnon mapping. Otherwise assume that fa = 1 and di = ei = 0 by sending to to the x-axis. The Jacobian condition shows that b 2 = 0. So G is of the form Note that bl 4= 0 and c a 4:0 are invariant under conjugations which do not introduce new terms. The different cases are listed below: ci(1 -- ca) = 0 yields B with b = d = b i = c2, c~4 = 1, b i=ca, 2aa+ ci(1 -- c~) 4= 0 yields A with b = b i ---- c~, c9 4 = 1, b l = ca, 2az + yields B with b ---- ba, d = ca, ca 4: 1, bi4:cz, b i4:1 ca4: 1, hi= I, al(l-- ca) a + aa(1 -- ca) + a~ = 0 yields B with b= 1, d=ca, HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I 11 cz4: 1, b 1= 1, ax(1--e2) 2+az(1-c2) +a~4=0yields C with d = cs, cs= 1, a s4=0yieldsDwithb=bl, c s= 1, a s-~O, b a4= lyieldsBwithb=bx, d= 1, cs= 1, a s=O, b 1= 1, 4ax--c~=OyieldsBwithb=d= 1, cs= 1, a s=O, bl= 1, 4a t-c~4=OyieldsCwithd~- 1. [] Remarks. -- The H6non family of mappings can be written in different forms. For example, Hdnon ([H~I] and [H62]) actually studied the family = ~x , ~ 4= 0. Note that Fa., is conjugate to H_,. _a. Thus, mappings of the form Fa, 0 were omitted in this other form. A fixed line is a line which is mapped onto itself (but not necessarily pointwise). Consider the set of lines y = k for all k ~ C. Elementary mappings can be understood by how they map these lines: all lines fixed, a unique fixed line, or no fixed line. A fixed line may be fixed pointwise, or there may be a unique fixed point or no fixed point. 3. Solenoidal mappings This section gives a classification up to conjugacy of unbraided solenoidal mappings, v:T ~ T of degree d, satisfying appropriate expansion properties and topological conditions. Solenoidal mappings, which are defined below, are injective mappings of degree d/> 2 of the solid torus. The images of such mappings can be braided and quite complicated. We only understand how to classify those which are unbraided. We will show that up to conjugacy, such mappings, when they are appropriately expanding and contracting, are classified by an integer. Only one of these mappings seems relevant to the study of H6non mappings. On the other hand, the authors puzzled about these mappings quite a bit while understanding the structure of H6non mappings, and we feel that it will be clearer if we study them all, if only to contrast the relevant one to the others. Theorem 3.1 holds for arbitrary mappings of degree d while Propositions 3.3, 3.5, and 3.6 require the mappings to be solenoidal. The construction of solenoids is given before Proposition 3.6. Proposition 3.7 shows that solenoidal mappings of degree 2 are unbraided while Proposition 3.8 requires unbraidedness. Theorem 3.11 is the classi- fication of conjugacy classes. Theorem 3.11 reduces the determination of a conjugacy class to the computation of an isotopy class and the verification of a hyperbolieity condition. 12 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Solenoidal mappings. Let D be the disk of radius 2, T = S 1 � D, and denote by (E, z) the coordinates in T. Definition. -- Let C+ and C_ denote the constant families of cones C+(~, z) ={(~,u I l~l~> lu[} and C_(E, z) ={(g,.) I Igl~< I"1} in the tangent bundle of T. Definition. -- A solenoidal mapping 9 : T -+ T of degree d is an injective C 1 immer- sion of degree d, such that, for all (E, z) e T and for some constant K > I, d,~,,, ~(c+(E, z)) c c+(<E, z)), (g, u) ~ C+(~, z) and dc~,, , -r(~, u) = (gx, ux) imply [gx [ > K IT [ and (g, u) e C_(E, z) and d,;,,, ~(~, u) = (gx, ux) imply [ u x [ < K [ u [. Remark. -- The definition says roughly that the derivatives of a solenoidal mapping preserve the family of cones G+ and are expanding in the g direction and contracting in the u direction in C+. From the fact that 9 is an immersion it follows that the inverses of the derivatives of a solenoidal mapping preserve the family of cones C_. Examples. -- Let S 1 --{E Cl I E I = 1 ),D ----{ zECl[ z ]~< 2},andT =S 1 � D. Define e 1 = { 1 } � 0 D and e~ = $1� { 2 ), each oriented by the counterclockwise orientation of the circle. We will examine very closely the following mappings, ~,,: T -+ T, which are unbraided solenoidal for every integer d t> 2 and k ~ Z: E d The reason for the shift in the exponent will become clear later: "~d, 0 has much nicer properties than the others. Theorem 3.1. -- For every mapping f: T -+ T of degree d >>. 2, there exists exactly d -- 1 continuous functions ~ : T ~ R/Z of degree 1 such that the following diagram commutes: T >T R/Z > R/Z For any two such mappings ~1 and ~z, there exists o with co a- 1 = 1 and ~x :- o~2. Proof. -- By the Lefschetz Fixed Point Theorem ([D]), f has 1 -- d fixed points counted with multiplicity. Since 1 --d 4= 0, there is at least one fixed point t 0. To avoid difficulties with branches of d-th roots, lift f to f~: T ~ 1", where HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I 13 = R � D is the universal covering space of T, with base point t o and the liftff: T -+ T satisfying 3~0) = ~0. Let y: ~"--> 'F be a generator of the fundamental group. Then a~V(7))---= y~ for every 7 e ~. Consider the space II = { ~ : "~ -+ R [ ~ is continuous and ~(y(t)) = ~(t) + 1 } with the uniform metric, 8, which is well-defined because of the periodicity. In this metric II is a complete metric space. Define a mapping q~ : II ~ 11 by ,(~) (7) = ~(f(7)). If ~ satisfies ~(ff(~')) = ~(~') + 1, then so does ~(~). Lemma 3.9.. -- The mapping ~ is strongly contracting. Proof. -- If ~1, ~ e II, then ... 1 1 ~ ~ I ~(~1) (7) -- ~(~) (7) 1 = 2 ] ~1(Y(7)) -- ~(f(7)) [ ~ d 8(=1, ~). [] (Lemma 3.2) Let ~o be the fixed point of ~ and n o be the mapping T ---> R/Z induced by ~o. Clearly the mappings rc k = e z'a~m-l~ % still semi-conjugate f to z ~-* z a. If rd : T -+ R/Z is any mapping making the diagram T >T ze-> z d R/z , R/z commute, then ='(to) is a fixed point of z F-+ z ~, so it must be one of the (d -- 1)-th roots of 1 and there exists k with ='(t0) --= =k(t0). Now r~' --- %, since the lift of e -2~/~a-~ re' in H is a fixed point of % hence is 7:o. [] (Theorem 3.1) Proposition 3.3. -- If a mapping f of degree d is solenoidal, then the mappings = : T -+ R/Z in Theorem 3.1 are fibrations with fibers homeomorphic to disks. Proof. -- Define H 0 C H, the family of IApshitz fibrations consisting of those ele- ments of H whose fibers are disks which are graphs of Lipshitz functions cr : D --> R (i.e. ] ~(zl) -- ~(z~) I ~< [zz -- z~ I)" Sinceff -x preserves the family of cones e_, the family of IApshitz fibrations is stable under ~. So the fixed point =o is a limit of IApshitz fibrations. The space II 0 is not closed in II, but the fibers of= e II 0 are fairly easy to understand. Lemma 3.4. -- For any rc = lim rc~, with rcj e IIo, and for any x eR, there exist two Lipshitzfunctions o~(z) <. %(z) such that =-~(x) = {(y, z) [ ~(z) ~<y ~< ~,(z)}. I4 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proof. -- We have = n u n , ;l(txo - xo + 9 >0 i>o j~>i Decreasing intersections of sets of the form r~-l(x) = {(y, z) I ~l(z) ~< y ~< 0c2(z)} are still of this form. Hence it is enough to show that U n ;l([xo - Xo + is of this form. Now suppose that r~-~([x0 -- ~, x o + e]) ={(y, z) ] %.~(z) ~<y~< ~j.~(z)}. Then U,>o o - Xo + = {(y, z) [ lim_.soou p. ~, ~(z) ~< y ~< lim,_.~oinf ~, ,(z)}. We need to show that %,~(z) ~< ~,~(z) for all j and ~ sufficiently large. For any fixed z, the function Y~-*~o(Y- z) is surjective, so there exists ~o with ~o(Y'o, z) = x 0. Choose I so large that 1~ -- ~o t < e forj >t i. Then ~j(y%, z) e [x o -- e, Xo + ~]. So for all j >i i, %.,(z) ~<Yo~ ~,,(z). [] (Lemma 3.4) Now if a fiber is not a Lipshitz disk, then it has nonempty interior. Also the fibers are compact and their projections onto R have bounded length. If any two points (xx, za) and (x2, zz) satisfy I z2 -- z~ ] < x 2 -- x~ and (x~, z~) =j~x,, z~), then x' 2 -- x'l >i K(xz -- xa) for some K > 1. If a fiber is not a Lipshitz disk, then let (xx, zx) and (x~, %) be two points of the interior satisfying [z 2 --z x 1< x~ -- x 1. Let ~,f""', z~"')=ff~ z~). These are still in the same fiber and x~- ~"~ -- x~ "1 i> K"(x z -- xx) for some K > 1. This contradicts that the length of the projection onto R of the fiber is finite. [] (Proposition 3.3) Proposition 3.5. -- The components 0ff~ n r:-l(z) have diameters tending to 0 as k -+ oo for all z ~ S 1. Proof. -- Suppose that x,y ~fk(T) are in the same fiber and realize the maximal vertical diameter, d~, offk(T). Compare this with the maximal vertical diameter, dk_l, of f k- I(T). Let xx ----- f- 1 (x) and Yl = f- 1 (y) and let t be the straight line joining them. The sdenoid N d. Given a space X with a mapping f: X-+ X, consider the projective limit Xt=lim(X,f) ={(...,x~,xl, xo) lf(x,+l) -=x i for i= 0,1,2,...}. When the mapping f is clear, simply write X instead of Xt" This construction is some- times referred to as the inverse limit construction. A point of this projective limit is a point of x 0 e X along with a " history of the point " under the iteration off HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I 15 The mapping f induces f: ~ -+ X by f(. . ., x,, xl, Xo) = (. . .,f(x2),f(xi),f(Xo)) -= (..., Xl, Xo,f(Xo) ) which is always bijective as f-l(..., xl, x0) = (..., xl). Consider the projective limit of the mapping 8:Si---~ S i given by ~(~)= ~a. Define Z a ----ljm(S i, 8) and the bijection ~:Z a -+ Z a as in the introduction above. This construction was studied carefully by Williams ([W]). The solenoid was first studied by Vietoris ([V]) and van Danzig ([vD]). Let r 9 be one of the d -- 1 mappings guaranteed by Theorem 3.1. Proposition 3.6. -- Let Z I = ~,f~ The mapping X v-~ (..., rr176 rcl(f-l(x)), x1(x), =i(f(x))) is a homeomorphism h~ : Xf ~ X a. Proof. -- Let z = (..., zl, z0) be a point of Z, and define X,, k = {x el" [f-'(x) is defined and ~/f-'(x)) = z, for 0,< i,< k}. Then fok maps ,:Ti(zk) bijectively to X,,~. In particular, Xz, k is a component of fo~(-~) n ~TX(z0). Since these components have diameters tending to 0 as k ~ 0% and h (z) = fl Xz, , we see that h~. is bijective. It is continuous, and the domain is compact, so it is a homeo- morphism. [] Definition. -- An injective mapping v : T ~ T of degree d is unbraided if there exists a fiber homeomorphism q~ : T -+ T such that @ o -: sends the core circle S ~ X { 0 } into S 1 � S i as a (d, 1)-torus (un)knot. Fro. 3.1 16 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Remark. -- No embedding of T in S 3 is specified. In particular, the (d, 1)-torus (un)knot and (d, d + 1)-torus knot are equivalent from this point of view. Proposition 3.7 shows that ff d = 2, all solenoidal mappings are unbraided; this is false for d t> 3. See Figure 3.1. Proposition 3.7. -- For degree d = 2, all solenoidal mappings are unbraided. Proof. -- We will show that there exists a fiber homeomorphism g: (T,f(S z X {0})) -+ (T, %,k(S 1 � {0 })). Note that the pair of sets (T, %,k(S 1 � { 0 })) is independent of k. Both (T,f(S t � (0))) and (T,-r ~ � (0))) are locally trivial fiber bundles over S 1 with fibers homeomorphic to disks with two marked points. Locally trivial fiber bundles over S 1 are classified by the isotopy classes of their monodromy. For the bundles under consideration the monodromy homeomorphisms lie in Womeo (D, { a, b )). The mapping M'omeo (D, { a, b }) -> ~erm { a, b } (the latter being the symmetric group on two elements) is surjective with contractible fibers ([Ha]). So the isotopy class of the monodromy of these bundles depends only on how they permute the boundary components. For both bundles, the points are exchanged. So the bundles are fiber homeo- morphic. [] Proposition 3.8. -- For every unbraided solenoidal mapping f there exists a unique integer k and a mapping h:T- int(f(T)) -+T- int(-~d,,(T)) suck that the following diagram com- mutes: OT > OT (3.9) 0f(T) > ~d,,(T) Proof. -- Step 1. There exists a fiber homeomorphism g : T -- int(f(T)) -+ T -- int(va, k(T)) mapping 0T to ST. Note that the set T --int(vd,,(T)) is independent of k. The definition of unbraided says that the bundles of pairs (T,f(S 1 � { 0 })) and (T, vd,,(S 1 � { 0))) are fiber-homeomorphic. Pick a base-point in S 1, and let D be the fiber above that base-point, { at,..., aa} = D nf(S1 � {0 }), with the points ordered along the circle f(S 1 � {bl, ...,bd}=D c~%.k(S 1 X{0}) ordered similarly. Then the bundles of pairs above are classified by their monodromies m I and ma. k. The definition of unbraided says that there exists a homeomorphism q~ : D --> D with c?({ al, ..., aa }) = { bl, ..., b a } and conjugating m s to md. k. Let U1,..., Ud be the components of D nf(T), labeled so that a~ ~U~, and Vt, ..., V a be the components of D n va.k(T), labeled so that b~ ~V~. Now deform 9 HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I so that 9(U~) = V~. This is (unpleasant) 2-dimensional topology. First adjust ~ in the definition ofzd, ~ so that $(U~) c~V~ = 0 for issj. Therefore the sets Vi II $(V~) have disjoint neighborhoods D~ homeomorphic to disks. Lemma 3.10. -- If U and V are closed subsets of the open unit disk D, with 0 e (J c~ ~r and both homeomorphic to closed disks, then there exists a homeomorphism ~ : D ~ D which is the identity on 0 D and with ~b(U) = V. Proof. -- Use conformal mapping to represent both D -- U and D -- V as stan- dard annuli, giving a system of" polar coordinates " where the radial curves are labeled by the points at which they intersect 0 D and the circular curves by their relative distance to 0 D. Then making points with the same coordinates in D -- U and D -- V correspond gives a homeomorphism of D -- l~l onto V -- V. This can be continued to U and V since any homeomorphism of the boundary of a disk extends to a homeomorphism of the interior, by radial extension, for instance. [] (Lemma 3.10) Find a homeomorphism d~: (D,{ bx,..., ba})-+V such that + o q~ is isotopic to ~p and + o ~p(U~)= V~. Unfortunately, this mapping does not now conjugate the monodromies, but it does up to isotopy, and that is enough, since bundles are dassified by the isotopy classes of their monodromy. Step 2. Next it will be shown that k can be chosen so that diagram (3.9) commutes on the level of homology. The homology group HI(0T ) is isomorphic to Z 2. Choose the basis { S 1 � { 2 }, { 1 } x 2S x }, the circles oriented counterclockwise in C. Consider the mapping g of- x o g- 1 o T 0. This is a fiber homeomorphism 0T -+ ~I', hence induces a mapping given by a matrix (I 0 i) for some integer t on HI(0T). Since the construction of g is unique up to isotopy, t is an invariant off. Observe that ~a,, can be written -ra, 0 o w ~, where w is the twist mapping w(L = So g of- 1 o g- 1 o "ra, k = g of- 1 o g- 1 o "ra, o o w ~. Since w ~ induces the mapping given bythematrix (; kl) on Hl(OT), set k= --t so that gof-log-~o~d, k induces the identity on the homology. Step 8. Finally, adjust g into h so that the desired diagram commutes. There exists a homotopy G~:0T X I~T with Go=.r~ogofog -1 and G~ = Id. Let U C T- int('ra, k(T)) be a narrow thickening of 0T homeomorphic to I � OT. Denote points in U by (t, x). Let 'g(y), y Cg-a(U) h(y) = G,(x), y eg-a(U) so that g(y) = (t, x) 3 18 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH on T -- int(f(T)). Now for y ~ ST, set x = g(y) and compute (f-loh-1 va,~oh ) (y) =f-~og-io.ca,~o (z~,~ og 0 --1 ofo g-~) (x) : g-l(x) :y. [] (Proposition 3.8) Next comes the classification of the conjugacy classes of unbraided solenoidal mappings. Theorem ~.11. -- Every unbraided solenoidal mapping is conjugate to one of the va, k, and no two of these are conjugate. Proof. -- The second part was proved above, when it was shown that different values of k lead to different values of l, which are conjugacy invariants. We wish to extend h from Proposition 3.8 to T. Take x ~ T. If x E X:, then define h(x) = ~a:k o ~: ~ X~a,k and if x r X:, then define h(x) = Va,~ o g :, where m is such that f~ int(f(T)). If f ~ 0f(T) so that f~ 0T, then both choices, m and m + 1, give the same value of h by Proposition 3.8. So the mapping is well-defined, bijective, and conjugatesfto va, k" It remains to show that h is continuous on Z/. Take x 0 ~ X:. The sets U~, ~(x0) = { x ~ T ]fo-.(x) exists for all n ~< N and I ~f(fo- ~(x)) -- ~s(f ~ ) I <~ ~ for n ~< N } form a basis of closed neighborhoods of x 0 as r -+ 0 and N -+ ~. Clearly k maps this basis of neighborhoods of x 0 to the corresponding basis of neighborhoods of f(xo). So h is continuous and hence a homeomorphism. [] 4. Embeddings of the solenoid in S a In this section we will try to describe the inductive limit of T under -% 0. Intui- tively, this corresponds to taking a solid torus winding around d times in a larger torus which winds around d times in a yet larger torus, etc., and taking the increasing union. This intuitive picture is ambiguous. To make this precise, the embedding mapping each torus into the next must be specified. This is made precise in this section. Smale first studied solenoids as hyperbolic attractors in S 3 ([Sm]). Recall the mappings ~a,, from the example near the beginning of section 3. Proposition 4.1. -- The mappings %,o extend to orientation-preserving homeomorphisms h a : S a -~ S a. Remark. -- Note that *a,k obviously extends to S a for some k. After all, one can take a solid torus (think of a bicycle tire tube) and wrap it d times around itself. The outside of the unwound tube and of the wound tube are both unknotted tori, so there exists a homeomorphism between them. This homeomorphism will map the inner rim ~oa,~-,~tx~ HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I of the tube to some curve on the wound tube; the object of this proposition is to describe this curve. The skeptical reader might experiment with a tube for d = 2, 3. The proof of Proposition 4.1 depends on the following: Lemma 4.2. -- If T 1 and T~ are two solid tori, and f: OT 1 -+ OT~ is a homeomorphism which sends curves on OT 1 which bound disks in T 1 into curves which bound disks in T~, then f extends to a homeomorphism T a -+ T~. Proof of Lemma 4.2. -- We may suppose T 1 = T 2 = T = S 1 � D. The homeo- morphisms of a torus are classified up to isotopy by their action on 1-dimensional homo- logy. If a homeomorphism of 0T extends to T, then any isotopic homeomorphism extends also. Clearly the linear homeomorphisms mapping curves of the form { ~ } � 0 D extend. [] (Lemma 4.2) 1/2~0 FxG. 4.1 Proof of Proposition 4.1. -- The key point is that ~a,o maps curves on 0T which bound embedded disks in S 3 -- T into curves which bound embedded disks in S 3 -- "ra, 0(T). This can be seen in Figure 4.1. This is a drawing of S 1 x D, with the disks { 1 } x D, { i } X D, { -- 1 } � D, { -- i ) x D; the reader is expected to fill in the other slices. Within these disks are d subdisks. The case d = 3 is represented, and the triangle formed by these three subdisks rotates by 1/3 of a turn while going around S 1 once. Thus these subdisks represent -rd, k(T). JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTI-I The curves drawn on the outside of the disks represent a disk X in S 8 -- "rd, k(T). Verifying that this is indeed a disk is the essence of the proof. We leave to the reader to verify that X is a manifold with boundary OX C 0~d,~(T). To see that this manifold is simply connected, notice that it is clearly a deformation retract of the subset consisting of X n S 8 -- int(S 2 -- T), and the star above -- 1. This is a contractible set: d disks, each with a leash and all leashes connected at one point, as in Figure 4.2. Now going around S 1 once, the angle at which X touches a subdisk rotates by -- (d- 1)/d, so that altogether 0X is the curve e ~'~t ~ (e 2d'n', e~,et + ce~im-d)). The mapping ~d,0 maps e~ (a curve bounding in S 3 -- T) to this curve, so ~a,0 extends as required, by Lemma 4.2. [] (Proposition 4.1) Fxo. 4.2 Reflections with respect to a torus. A different way of understanding the exten- sion of ~a,0 to S s will be given requiring a definition of reflection with respect to a torus. The simplest context in which to describe such reflections is to write S =={(u,v) l lul 2+1 vl 2= 1}. Then S s = T' to T", where T'={(u,v)~S'llulal/V'2} and T"={(u,v)~S31lvl<~l/V~}. These are two unknotted solid tori, and dearly they are exchanged by the mapping HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I 21 To give an intuitive description, we will work in S ~ = R 3 t3 { dO }. So this mapping needs to be translated into a mapping R a t3( do )~ R 3 u{ dO }. Stereographic pro- jection from the point (0, i) maps S ~ to R 3 ~3 do according to the formula Re u/(1 -- Im v)) (~)~ Imu/(1--Imv) . Re v/(1 -Im v) This mapping takes the torus l u [ = e~~ Iv [ = e~~ to the parametrized torus in R a /cos 01/(v - sin (01)0 v'-'/sin 01/(3/2 --sin 0~)~, \cos 0~/(A/~ -- sin 0~) ] which just happens to be the torus of revolution obtained by rotating the circle of radius 1 centered at (We2, O) in the (x, z)-plane around the z-axis. Conjugated by this change of variables, the mapping p becomes ( z ) x ~+ (y-- 1) ~+z ~ x~ + y~ + z2- 1 . 2x Note that Pl commutes with reflection in the y-axis. Fro. 4.3 Construction of h a. Consider two unknotted solid tori T O and T 1 embedded in R a, linked with linking number d, as in Figure 4.3. Then R d : S 3 -+ S 8, the rotation by 7rid around the z-axis, is a homeomorphism of each onto the other. For any homeomorphism ~ : S 3 -+ S 8 set p~ = ~-1 o Pl o ~. JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proposition 4.3. -- There exists an orientation-preserving homeomorphism such that a) ~ maps T O to T'; b) ~ commutes with reflection in they-axis; c) the restriction of h a : p~ o R a to T o is a solenoidal mapping conjugate to ~Va, o; d) h a is conjugate to its inverse. Proof. -- Fiber both T O and T1 over the circle by the radial angles, as measured from they-axis, and similarly for T'. Choose first the restriction of the homeomorphism to To, so as to map the slice with a given radial angle of To to the corresponding disk of T' and so that ~ commutes with symmetry with respect to the y-axis. Next choose a curve 7 winding d times around T', symmetric with respect to the y-axis and such that the " radial angle " of T" is monotone along the curve, and a small tubular neighborhood S around it. Note that this radial angle of T" will increase by 2dn along ~. Fiber S by the radial angle, starting at the highest intersection on the y-axis. See Figure 4.4. Fio. 4.4 Next, define 0~ on T 1 by sending the slice at a given angle to the slice of S at d times that angle, still preserving the symmetry with respect to the y-axis. Extend the homeomorphism to S 3 so as to preserve the symmetry. With this choice of ~, a) and b) are clearly true. All the work was designed to satisfy c) and d): the restriction of ha simply multiplies radial angles by d in To, hence is expanding in that direction. By choosing the tubular neighborhood S of y sufficiently thin, ha can clearly be made contracting in the slices. Since y is unbraided in T", the sole- noidal mapping ha : To -+ To is conjugate to va, k for some k, which must be 0 since ha extends to S 3. The inverse of ha is Ralo p,, which is conjugate to p~o Ra 1. Conjugate the mapping by symmetry around the y-axis. This conjugates R a to R~ -1, and since the HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I reflection with respect to the y-axis commutes with p,, R o P0 is conjugated to its inverse. [] This shows that the mapping ha : S 3 -+ S 3 is a homeomorphism, which has two invariant solenoids Z+ C T O and Y,_ C T1, attracting and repelling, respectively; every point is attracted to ~+ under forward iteration of ha and is attracted to Z_ under itera- tion of h~ -1. haduetlve llmlts. Given a space X and a mapping f: X -> X, define the inductive limit lim(X,f) to be li_m(X,f) = X x N[,~, where the equivalence is generated by setting (x, m) ,~, (f(x), m + 1). The notion of inductive limit is pathological when f is not injective (the spaces created fail to be Hausdorff). We will use the notion only for injective mappings f, where it really is some sort of increasing union. Proposition 4.4. -- The inductive limit lira(T,-cd,0) is homeomorphic to S 8- Z a and r.a = N.. ~2,o(T). Proof. -- The mapping (x, m) F-. k~-'(x) induces a mapping lim(T, ~a, 0) -->Um ha re(T0). The mapping ha is conjugate to its inverse, and the conjugating mapping is a homeo- morphism of S ~ -- h~-re(T0) onto ha(T0). [] Corollary 4.5. -- The fundamental group nl(S 3 -- Za) is isomorphic to the additive group Z[1/d] of rational numbers witk powers old in the denominator. Proof. -- Fundamental groups commute with inductive limits, so, by Proposition 4.4, nl(S s -- Za) is isomorphic to the inductive limit of zLzLz...o Remark. -- It is usually dangerous to speak of fundamental groups without spe- cifying a base point, but in this case the fundamental group is abelian, so there is no ambiguity. Knots and the mappings ~d,,. We Mll not need the following results in the sequel, but they may help the reader to understand why the mappings ~a,, are different. We will only discuss the case d = 2, but a similar discussion can be made for arbitrary d, and is a bit simpler in fact when d > 2. 24 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proposition 4.6. -- The solid tori "r~,~(T) are all unknotted if k = 0 and all knotted for n >i 2 if k # O, except that ~, _a(T) is unknotted. Proof. -- The case k = 0 is dealt with in Proposition 4.1. Next show that v~,~(T) is the (2, 2k -- 1) torus knot. This is genuinely knotted unless k = 0 or 1. Since ~"+~rT~,k ~ j is a companion of ~,k(T), this proves the result for all k except k = -- 1, which requires a separate argument. Observe that v~,k can be written v2,0 o w k, where w is the twist w(~, z) = (~, ~z). Then x~,~(T)= "r2,0ow~o'r2,0owk(T). The w ~ on the right can be ignored since T = w~(T) and since v2,0 extends to a homeomorphism of S a, the v~, o on the left can be ignored also. The result follows from the computation The mapping ~ ~-~ (~z, ~2k+1) is a parametrization of the (2, 2k + 1)-torus knot, which is indeed knotted unless 2k § 1 = 4- 1. To finish the proof, it must be shown that v~,_I(T ) is knotted. As above, "r~,_l(T ) = vz, o o w -1 o "r2, o o w -1 o "r2, 0 o w-I(T), and again ignore the w- ~ on the right and the %, 0 on the left. The reader may check that the core of the solid torus is then parametrized by ~ ~-~ (~4, ~-2 q_ r We leave it to the reader that this is a parametrization of the (2, -- 5)-torus knot. [] 5. The functions G mad q~ Recall the generalized H6non mappings of degree d, where a # 0 and the degree of q is less than d. Recall also the definitions of sets K and U i from the introduction. Looking at the formula for the H6non mappings, note that if x is reasonably large and large with respect to y, then the predominant behavior is that the x-coordinate gets raised to the d-th power. The following definitions are designed to state this rigo- rously. Set ~ to be at least as large as the absolute value of the largest root of Ix I - I q(x)l - (I a I + 2) x = 0. If p(x) = x2q - c, then the following value of ~ works: ~=~([a] q-2q-%/([a I +2) 2q-4[c[). HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I Define the regions V+, V_, and W C G ~ by V+ =((;)[ lyl~< Ix] and Ix[~> ~z}, w = lxl.< ~ and lyl.< ~ and Lemma 5.1. -- The sets V+ and V_ have the following properties: a) V+ C U and U+ = [.J,>~o F~ U_ = [-J.~>o F~ b) r(v+) cV+, F-I(V_) CV_; c) if (x,y) eV+, then [(F~ 2" for n= 1,2,... and if (x,y) eV_, then 1( F~ 1 + as n= 1,2,...; d) F(W) c W u V+; e) if (&y) eV_, then I F,(x,y)l < lYl and if (x,y) eV+, then I F~-a(x,y)l < Ix I" Proof. -- First consider the statements for V+. To see part b) let (x,y) ~ V+, and calculate: [p(x)-ay >>. ]p(x)]--]a[lyl>~ [p(x)]--[a[[x[>~ 2[x[. Thus F(V+) C V+ and for all n = 1, 2, ... Hence part r and also part a). Part d) is obvious since F(W)C{(x,y)l ly[< e}. Part e) is obvious also. For (x,y) ~ V_, the proofs are analogous using 1 1 [ a~l I P(Y) -- x [ >/~ I(P(Y) I - I x l) />~(IP(y)I-lyl)>~ 1+ lyl. [] Remark. ~ The proof of part c) of Lemma 5.1 shows that the first coordinates of an orbit starting in V+ grow at least geometrically. This is actually misleading. Since the dominant term of F1 is of degree d, the growth is like k a", Lemma 5.1 shows that every point eventually lands in V+ u W. 4 26 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proposition 5.2. -- There exist unique analytic functions q~+ : V 13- b suck that ep+ (;) -- oo in V+ or V_, respectively. and Proof. -- The function q~+ is constructed below and shown to have the required property. The proof for ~?_ is analogous. The proof of uniqueness will be given at the end of section 8. To simplify the formulas below set the notation x, = (F~ (x,y) and y, = (F~ (x,y). Note that x, is a polynomial in x and y of degree d" whose sole leading term is x a" and y, is a polynomial in x and y of degree d "- ~ whose sole leading term is x a"-~. To define q~+, meaning must be given to the limit ;) lim x lldn or rather the equivalent telescoping infinite product (;) xl/d lid n+l X . "1 Xn + l cO+ x ..... x~ an " " " Examine the individual terms of this infinite product: X~'l+11/dn+l -- [4 -~- q(x.) -- ayn] llan+l xlld n x~d n For (x,y) ~V+, F~ belongs to V+ and I q(x) - ay I q(x) l + l ly I ~< I q(x) l + I a II x I Ix I a -- 2 Ix ] ~< Ix Ix F <~ Ix I x 2 2 = 1 ixF_l~< 1 -- ~ -. Now, for the d"-th root use the principal branch of (1 + z) va". The infinite product converges as the series of the logarithms of the terms in the product converges. In the product above, consider the factor HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I The terms of highest degree in both polynomials involve only x. Since ]y [ ~< [ x [, the term (q(x,) -- ay,)/xa, is of order 1/x a". That is, there exists a constant C such that in V+ q(x,) -- ay, C which tends to 0 as (x,y) -+ oo in V+. Therefore the product is equivalent to its first term x. [] A refinement of this result will be needed, pushing the asymptotic development of ?:~ a bit further. We find it easiest to write = Uo(X) + ul(x)y + .... as a convergent power series in y, with coefficients Laurent series in x, which is clearly possible by the structure of V+. Proposition 5.4. ~ The following asymptotic, development holds: Uo(X) = X + O(Ixl) ana u (x) = dx _ 1+ o Proof. -- The development of u 0 is already in Proposition 5.2. From (5.3) above, the second and higher factors of the product cannot contribute larger terms than those given, and the first term gives the result. [] Proposition 5.5. -- The limits exist, are continuous on C ~', are pluri-harmonic on U + , and have the properties that Moreover, Proof. -- Again, the proof will be given for G+ and the proof for G_ is analogous. On V+, define G+ = log ] ~+ ]. Extend this definition to (x,y) ~ U+ as follows. By part a) of Lemma 5.1, there exists n > 0 such that F~ e V+. For such (x,y), define Further extend G+ to be zero on K+. 28 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH The definition is consistent: if a higher n had been used, then the result would be the same by Proposition 5.2 and clearly satisfies G+(F(x,y)) = dG+(x,y). The function G+ is harmonic on U+ since it is a real part of an analytic function on V+ and elsewhere the pullback of a pluri-harmonic function by an analytic mapping. It remains to see that it is continuous. Fix (x',y') ~J+. Then there exists N such that [[F~ ~ for all n/> N. For any M>N, there exists ~ so that if l[ (x",y") -- (x',y')II < =, then II F~ '') I[ < 0~. Note that F(W+) C W+ to V+. So the value of G+ on the first forward image of F~ '') which is in V+ is bounded by C = sup{G+(F(x,y))l(x,y ) eW}. So, G+(x",y")< C/d ~. [] Remark. -- The functions G~_ are obviously subharmonic. This fact has been observed by Bedford and Smillie ([BS2]) and by Fornaess and Sibony ([FS]). They use the fact that dd ~ G+ are positive (1, 1)-currents supported on J to derive analogs of the Brolin measure ([Br]) for Hdnon mappings. Fornaess and Sibony also prove that G are H61der continuous. 6. The global topology of H~non mappings The behavior of G+ is partially described by the following, in which solenoids make their first appearance in this subject. Theorem 6.1. -- The mapping G+ : U+ -+ R+ is a trivial fibration whose fibers are homeo- morph# to S 8 -- Y"a, o, embedded using the mapping "ca, o as in section 3. Proof. -- Represent the set U+(r) = G+l(log r) as the increasing union U+(r) = V+(r) to F-l(V+(r~)) to F-z(V+(r4)) to..., v+,., Jo+(;) / Proposition 6.2. -- a) For large s, V+ (s) is homeomorphic to a solid torus, and : V+(s) --+ { I I * I = s ) is a fibration with fibers homeomorphic to closed disks. b) The mapping G+ : V+ -+ R+ is a trivial bundle with fibers homeomorphic to solid tori above (R, oo)for R suffciently large. Proof of Proposition 6.2. -- For any z with ]zl~< 1 consider the function %(x) = 9+(x, zx). The function is defined and analytic for Ix[> 0r By Proposition 5.2, % has a simple pole at oo. The following lemma, which is an immediate consequence of Montel's Theorem, will be required. HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I Lemma 6.3. -- Let R > 1 ; then the space of analytic functions f: D -+ D~, satisfying 3,.(0) : 0 and f'(O) : 1 # compact. In particular, there exist numbers R1 and R e such that all such functions f are injective on D~I , and satisfy f(DR1 ) D D~. Applying Lemma 6.3 to ~+ (l/x, z[x)' which maps the disk of radius 1/, < 1 to the disk of radius 1, we see that there exist R x and R~ such that if ~ > R 2 and [ z ] ~< 1, then there exists a unique x such that ] x I > Ri and ~0+(x, zx) = ~. This shows that the mapping  : (x,y) ~ (~+(x,y),y/x) is a homeo- morphism V+(r) -+S i, � D for r~> Ri. [] (Proposition 6.2) To compute F in the coordinates (~, z), asymptotic developments of x and y as functions of ~ and z must be found. Proposition 6.4. -- The following asymptot# development holds: Proof. -- This is a standard inversion of an analytic function from Proposi- tion 5.3. [] (Proposition 6.4) Now compute F in the coordinates (~, z): (:) . \zx(~, z)] ~ x(~, z) p(x(~, z)) - azx(~, z)) Only the term ~a in the denominator contributes to the leading terms of the development of F, to give r = 1 +o + +o z+o(Izl) This mapping is not quite one of the va,~'s from section 3, but almost. Change variables once more, to (~, ~), where ~ = z~. In these coordinates, the following expres- sion holds: (6.5) F(~)= +o(1~1)+ d~_i+o ~-t-o(l~l) " ( t t tt ) In particular, it is conjugate to vd, o. 30 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Consider 0V+(r) with r sufficiently large so that Lemma 5.2 applies. Let y, be the curve parametrized by t ~-* (~ = e 2~", ~ = ~rd), 0 ~< t ~< 1, and let ~ be the curve parametrized by t ~ (~ = r, ~ = 0or a e2~t), 0 ~< t ~< 1. Proposition 6.6. -- If g : V+(r) -+ T is a homeomorphism with g(y,) a curve on T bounding a disk in S 3 -- T, then there exists a homeomorphism g' : V+(r a) -+ T such that the diagram V+(r) o > T V+(r d) -----+ T g' commutes. Moreover, g'(y,d) is again a curve which bounds in S 3- T. Proof of Proposition 6.6. -- The existence of g' and its uniqueness up to homotopy is an unpleasant topological generality. The substance of the proposition is in what g' does to y,d. The generality is a consequence of the following lemma. Lemma 6.7. -- Let X be a 2-sphere with three open disks with disjoint closures removed. Then the space of homeomorphisms of X mapping each boundary component to itself is contractible. Proof of Lemma 6.7. -- See [EE] and [Ha]. [] (Lemma 6.7) Both V+(r a) --F(V+(r)) and T- Va.o(T) are locally trivial fiber bundles over the circle with fibers homeomorphic to the sphere with three holes above. In each case, the functions called ~ are the fibrations. The following shows that these two spaces are fiber-homeomorphic. Cut the circle at some point, to manufacture two bundles V+ (r) and "r of spheres with three holes over the interval I. Both are trivial bundles, and hence homeomorphic toI xX. ~,~ Choose trivializations v:I x X~V+(r) and u:I x X4T. These induce monodromy mappings m.= (ul{1}xx)-' o Ul~o}xx and m.= (VI{1}XX)--I 0 VI{O} XX. The mapping u o v -1 would induce the desired homeomorphism V+(r) --->T if m~ o m; -1 were the identity. To arrange this, let m t be a family of homeomorphisms of X such that m0 =m~-Zom~ and m 1-=Id and define m:I � X-->I X X by re(t, x) = (t, mr(x)). If u is replaced by the trivialization uj = u o m, then the require- ment is satisfied. This manufactures a homeomorphism w = uz o v-Z:V+(r d) -- F(V+(r)) ~T -- vd, o(T). HI~.NON MAPPINGS IN THE COMPLEX DOMAIN. I 31 It is clear from the construction that its isotopy is unique (among fibered homeomor- phisms). It must be shown that w can be adjusted so as to coincide on 8(F(V+(r))) with za, 0 o g o F- 1 and that w maps ~,r to a curve on T which bounds a disk in S a -- T. Both questions are homotopy class questions: the second one obviously and the first because the restriction of w to the boundary can be adjusted to coincide with any homeomor- phism in its homotopy class. Both of these statements follow from the asymptotic expansion (6.5): + ~ + ~(~, n) where the error term is so small that if a parameter is put in front of it and varied from 1 to 0, then no homotopy classes are changed. Once the parameter is 0, the formula looks exactly like the formula for zd, 0" This is slightly misleading since ~ and B are in the circle of radius r and the disk of radius r, respectively, whereas the arguments of Cd, 0 are in the circle of radius 1 and disk of radius 2, respectively. We leave it to the reader to make the appropriate scaling after which the identity is a possible candidate for w. [] (Proposition 6.6) The proof of Theorem 6.1 is completed by induction. The same construction as above gives a sequence of homeomorphisms g', g", ..., where g(k) : V+(rdk) _+ T. Define G(k):F-k(V+(rak))---~'r~,ok(T) by G(k) = ~-ka, o o g(k~ o F k. That is, the following diagram commutes: 17+ (r) r ~ F-l(V+(rd)) r ~.F-~(V+(rd2)) c ~F-s(V+(rdS)) c :.... F2 1 F3 G " ~ d~ G " F~ d s " ~V+(r ) ) ,... 9 r_~,) a,u- * T. c g'[ ;- r~o3(T) c ..... ~T ~T ~... rd, O r~0 r~o In the end, U+(r)= [.J~~ is homeomorphic to IJ ~ s ~ k=o X~,ok(T) = _ Za, 0. This proves that the fibers of G+ are homeomorphic to S 8- Za, 0 for r sufficiently large. Proposition 6.6 admits parameters: if gt were a family of homeomorphisms as in the proposition, depending on a parameter t in an interval, then there exists g~ depen- I,V+(rg.,~[ 32 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH ding continuously on t and satisfying the conditions of the proposition. In particular, there exist homeomorphisms g, g', g", ... such that the following diagram commutes: U,~>~V+(r) ~> U,>RdV+(r ) ... T X [R, oo) ,d,% T X [R, oo) ... Applying this extension of Proposition 6.6 to the inductive proof above shows that the mapping G+ : U+ -+ R+ is a trivial bundle above (R, oo), with the same R as in Proposition 6.2. Now F ~ is a fiber homeomorphism of G+I((R/2 k, oo)) to G+I((R, oo)), covering x ~ 2 k x. Thus the mapping G+ is a trivial fibration over any compact subset of R+, hence locally trivial over R+, hence trivial since R+ is contractible. [] (Theo- rem 6.1) 7. The foHations of U+ The fibers of G+ are 3-dimensional manifolds, and not obviously objects of complex analysis. But because G+ is a pluri-harmonic submersion, U+(s) is naturally foliated by Riemann surfaces. We will show that every leaf is isomorphic to C and dense in U+ (s). The proof also shows that q~+ cannot be extended to all of U+. Lemma 7.1. -- Let W be open in C" and let h : W ~ R be a pluri-harmonic submersion. Set W(x) = h-l(x). Then each W(x) is a real (2n -- 1)-dimensional manifold, and it is naturally foliated by complex manifolds of dimension n -- 1, with tangent space at w E W(x) given by T~ W(x) tn iT,, W(x). Proof. -- Each W(x) is a manifold by the Implicit Function Theorem. The uni- queness of the foliation follows from the fact that a real hyperplane T of a complex vector space contains a unique complex hyperplane, namely T t% iT. The existence can be seen by setting locally h ---- Ref for some complex analytic function f, which is also a submersion, and observing that W(x) =f-l({ z ] Re z ---- x}) is naturally foliated by the fibers of f, which are complex manifolds of dimen- sion n- 1. [] Theorem 7.2. -- The leaves of the natural foliation 0fU+(s) are isomorphic to C and each is dense in U+ (s). Proof. -- Choose ~ e C -- D with [ E I - s, with s so large that Proposition 6.2 applies. The leaf through any point of ~+1(~) can be written ?+1(~) u F-~(?+*(~a)) u F-~(?+l(~a")) v... By Proposition 6.2, this is an increasing union of simply-connected surfaces, hence simply connected. HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I 33 V+ (s) . v+ (~) ~+ 'I C-,C ~ Co FIo. 7.1 F ~ (v+(s~)) /- F -s(v+(ss)) IOOOQ'~ FIo. 7.2 To see that this leaf is dense in U+(s), note that F-I(~I(~)) = U ~+I(o~) o~d= 1 (see Figure 7) and more generally F o-,(~u = ~U ~u 1(~). Since the d*-th roots of 1 are dense in the unit circle, each leaf is dense in V+ (s). Applying F repeatedly will make it dense in each term of the increasing union V+(s) kjF-I(V+(xd)) k./F-2(V+($a2)) kJ ..., which occurred in the proof of Theorem 6. I. It remains to show that the leaves are isomorphic to C. This requires the following proposition. 5 34 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proposition 7.3. -- Let X be a simply connected Riemann surface, and K C X a compact connected simply connected subset not reduced to a point. Suppose A0, A1, . .. is a sequence of disjoint annuli in X -- K such that the inclusion of A~ into X -- K is of degree 1. If oo mod(A~) = 0% if0 then the surface X is isomorphic to C. Proof. ~ The alternative is that X is isomorphic to D and X -- D is an annulus with finite modulus M. However, by the subadditivity of moduli of disjoint homotopic annuli ([A], [BH]), ~E~~ 0 mod(A~) ~< M. [] (Proposition 7.3) So find a sequence of annuli in a leaf with a divergent series of moduli. This is actually easy, as the annuli considered grow very rapidly. Consider the annulus For ~ sufficiently large, the function y/x is an isomorphism of A; onto the annulus 1/2 ~< [ z [ ~< 1 of modulus (log 2)/(2rc). The annuli A~, F-a(A;d), F-~(A;d,), ... have constant moduli. They are embedded in the leaf which contains q~+l(~), disjoint by Lemma 5.1, and embedded with degree 1 in the leaf with ?+1(~) removed. So the leaf is isomorphic to C by Proposition 7.3 (see Figure 7.2). This proves the result for [~1= s sufficiently large. The statement follows in general by observing that F maps bijectively leaves in U+(s) onto leaves in U+(sd). [] (Theorem 7.2) Proposition 7.4. -- The mapping induces a bijection of the set of leaves onto the (non-Hausdorff) group R/Z[1/d]. Proof. -- This was already shown in the proof of Theorem 6.2. [] Remark. -- There are analogous results for those in Sections 7, 8, and 9 if" F-1 ,, replaces " F ". HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I 35 8. An analytic description of U+ In this section the analytic structure of U+ is analyzed completely. This is done by showing that the " Riemann surface of ~+ " is a covering space of U+ isomorphic to (C -- D) � C. Therefore U+ is a quotient of (C -- D) � C by some discrete group of automorphisms, isomorphic to Z [2]/Z. The group of automorphisms of (C -- D) � C is infinite-dimensional, and since the covering group we are after is only defined up to conjugation, there is a good deal of freedom in the description. The particular choice is algebraically very pleasant, but may not be the best one from a dynamical point of view. The Riem~nn surface of ?+. Let U+ be the smallest quotient of the universal covering space of U+ on which ~+ is defined. This covering space should be thought of as the Riemann surface of ~+, but it cannot be defined as a subset of U+ � C since the fiber above a point of U+ is a coset of the group of dyadic angles, and hence not discrete in C, so the topology would be wrong. Being a covering space of an analytic manifold, U+ is a 2-dimensional complex manifold. The set V+ is naturally embedded as an open subset of U+, using the natural definition of V+ on V+, and of course there is an analytic function ~+ : ~+ -+ C -- which extends ? on V+. This mapping ~+ is a submersion, and its fibers are simply connected Riemann surfaces, hence isomorphic: to D, C, or the Riemann sphere. Theorem 8.1. -- The fibers of ~+ are isomorphic to C. Proof. -- This follows from Theorem 7.2. [] It is unfortunately not true that a 2-dimensional complex manifold with a sub- mersion to a subset of C and with fibers isomorphic to C is a locally trivial family of copies of C. Example. -- Let U ---- D � ~, where ~ is the Riemann sphere. Choose some non- analytic continuous mapping ,:D-+ C, such as ,(z) ----2. Consider the set U = U -- (graph of ~). The projection U -+ D does have all fibers isomorphic to C, but if it were analytically a locally trivial fiber bundle, then the section ~ would be analytic. However, with an extra condition, such submersions are locally trivial fiber bundles. Let X, Y be complex manifolds, and f: Y ~ X an analytic submersion. Let the vertical tangent bundle Ty/x =ker dr. Recall that a vertical 1-form is a section of ~om(Ty/x, C). Proposition 8.2. -- If all the fibers off are isomorphic to C, and if Y carries an analytic vertical 1-form ~, such that the integral of ~ along a path in one fiber vanishes only if the path is closed, then the mapping f: Y ~ X is a locally trivial fiber bundle. 36 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proof. -- Choose any x ~ X, there exists a neighborhood U of x and a section ~: U ~ YoffNowdefineamappingg:f-l(U) -+ U � C bysendingy to (f(y), ~/~,, o~), where "t(Y) is a path in f-~(f(y)) joining r toy. There exists such a path since the fibers are connected, and the integral is independent of the choice since the fibers are simply connected, and all analytic 1-forms on a Riemann surface are closed. Clearly g makes the diagram f-'(u) uxc XZ , commute, and g is an isomorphism fiber by fiber. Indeed, the hypothesis implies that g is injecfive on each fiber, and an injective analytic mapping G ~ C is an isomorphism. [] Theorem 8.3. -- The projection ~+:U+ ~ G- D is a trivial analytic fiber bundle. Proof. -- By Cartan's Theorem B ([G]), it is enough to prove that it is locally trivial, since there are no topologically non-trivial affine-line bundles over C- D. Moreover, C -- D is a Stein domain, so the topological and the analytic classifications of such bundles coincide. Since log ?+ is well defined up to an additive constant, the 1-form co = d log ~+ is well defined on U+. Moreover, o~ has no zeros since any branch of log q~+ is a sub- mersion. Therefore one can locally find a function g on U+ such that dg A co --~ dx A dy. Let + be the restriction of ~'~_ dg to vertical tangent vectors. Since dg is well defined up a multiple of o~, this restriction gives a well defined vertical 1-form. To avoid conflict of notation with the exterior derivative set ~ = d in the following. Lemma 8.4. -- We have F* hb = (a/8) ~b. Proof. -- Clearly F*eo = 8o~, and F*dx ^ dy = a dx ^ dy. Thus up to multiples of co, F* dg= (a/~) dg. The result follows. [] (Lemma 8.4) Now to show that the criterion of Proposition 8.2 applies to +, project a curve in one fiber of U+ to U+. This projection will be closed only if the original curve was closed. Further take forward images of the curve until it lies in V+(r), for sufficiently large r. This will change the integral of + by dividing it by a power of a[8. So it is enough to show that for ~ sufficiently large, the integral over a curve y in q~+(~) vanishes only if 3" is closed. By Proposition 5.2, ~+ ,~ x, so that ,.~ dx/x, so that g can be chosen with dg ,-, x dy. Since the path y is nearly vertical, this term of dg contributes more than all other terms, and hence for such an integral to HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I 37 vanish, they-coordinates of the endpoints must agree. But this means that the endpoints agree, by Proposition 6.2. [] (Theorem 8.3) Next the structure of the group F C Aut U+ such that U+ = U+/r is examined. Proposition 8.5. -- The fibration ~J + ~ U+ induces an exact sequence of fundamental groups 0 , , 9 r , 0 >Z >Z Z >0 where I is represented by the canonical generator of hi(V+). Since V+ lifts to U+, this verifies that the left square is commutative, and the remainder follows. [] Proposition 8.6. -- There is a unique lift F ofF to ~J + mapping V + to V+, and it satisfies F(y(x)) ---- (8u F(x) for all y e F, where the composition law of F is written additively. Proof. -- Elementary covering space theory shows that the lift exists and is unique. The formula then comes from the fact that F : V+ --> V+ induces multiplication by on the fundamental groups. [] For the remainder of this section, let us restrict ourselves to degree 2, with p(x) = x ~ + c. It is possible to find similar formulas in higher degrees, but they require inverting a power series, and the computations are difficult and do not lead to simple expressions. Theorem 8.7. -- There exists a unique isomorphism ~J + ---> (C, -- I)) � G such that in that trivialization, the mapping F is written F(~, z) = ~2,~ z § ~. Proof. -- Choose a trivialization of the bundle U+ --> C -- D so that the zero section is tangent to a high order to the section s o : ~ ~ (x(~), O) at ~. There exists such a trivialization: in any trivialization, s o is a power series in ~ which converges in some neighborhood of oo. The sum s, of the first n terms of this series is an analytic section over all of C -- D and arbitrarily close to s o as n -+ oo. Now change trivializations so that s, becomes the zero section. Next an aymptotic expansion of F in this trivialization will be computed. 38 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Lemma 8.8. -- If n is sufficiently large, then ff has an asymptotic expansion z) = z + + o(1) . Proof. -- Using a trivialization such that dz corresponds to +, then by Lemma 8.4, F* dz ---- (a/2) dz. This means that ff will act on each fiber as z v-+ (a/2) z + constant, and we are left with computing the constant (which depends on the fiber, i.e., on ~, of course). This " constant " can be understood as follows: in the chosen trivialization, take the point s,(~), and integrate qb from s,(~ 2) to F(s,(~)) along a path in the fiber. Return to the definition of de above. It was found from a function g on V+ satisfying dlog ~?+ ^ dg -~ dx ^ dy; so this integral is just g(F(s,(~))) -- g(s,(~)). In the formula above, replace s, by So, and only change arbitrarily small terms in the asymptotic expansion. Since q~+ is to first order x, setting g(x,y) ~ xy satisfies the equation d log q~+ ^ dg = dx ^ dy to first order. We invite the reader to check that ignoring the other terms of g will not affect the asymptotic expansion above. Setting (x(~), 0) = s0(E), compute g(x(~) 2 q- c, x(~) ) -- g(x(~2), O) = (x(~)" q- c) x(~) q- terms to be neglected. In Proposition 6.4, we started to compute x(~), but dit not quite go to the required precision. In fact, it would have been quite difficult to extract the relevant terms of the 0(1 ~ I) in arbitrary degree. The goal is the following formula: x(;) = +o , which we leave to the reader to verify. Then g(x(~)*-q-c,x(~))--g(x(~*),O)= ~--~-~ q- ~--~ q-0(1) 2~ q- 0(1). [] (Lemma 8.8) To complete the proof of Theorem 8.7, the o(1) above must be dealt with. This is some function v(~) on 13- D which vanishes at infinity. Making a change of trivia- lization (~, z) ~ (~, z + u(~)), the expression of F in the new tHvialization is c a HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I Next find u(~) so that v(~) -- ~ u(~) + u(~ 2) = 0. This can be done by formal power series, or by setting Clearly this series formally solves the problem and it converges since I v(~ 2~') [ ~< C I ~ ] -2m for some constant C. This shows the existence of the required trivialization. The uniqueness is clear from the uniqueness to the solution for u above. [] (Theorem 8.7) Finally the group P may be computed. Theorem 8.9. -- For each element j/2 k e r, there exists a unique polynomial P~,k(~) suck that in the trivialization above, the element of r corresponding to j]2 ~ is given by + p~,~(~)/" Proof. -- First, compute 7v2. By Proposition 8.6, c 9 z+~-~; This leads immediately to "~1/~(~, z) = -- ~, z + a More generally, suppose that ~'j/~k-t has been determined. Then Proposition 8.6 gives z+ ~ which can be rewritten +pj,~(~) + (e2"J/zk~) s ~ z + c 2 " z _ c e2,~j/~k = ~8 _ ~ +Pj-I,~(~ )~ This gives us pj,~(~). [] Finally, we fill in a gap in section 5, the uniqueness of q~ in Proposition 5.2. This actually requires knowing Theorem 8.7 for H6non mappings of any degree; the proof goes through with minor changes. 40 JOHN H. HUBBARD AND RALPH W. OBERSTE-VOKTH Proof of uniqueness for ~+. -- Suppose ~b : V ---> C -- D satisfies (+([;])t and First observe that + lifts to ~ : ~+ -+ C -- D. This is an application of the lifting cri- teflon for covering spaces. For all n, the space ~, = F-"(V+) has the homotopy type of a circle, and F : V, ~ ~,_ 1 induces multiplication by d on the fundamental groups. Thus + can be lifted recursively to all ~, by the formula and the proper choice of roots will guarantee that all lifts agree with t~ on V+. On each fiber of ~+, the function ~' must be constant, since it is a mapping C ~ C -- D. Thus we can write q~ = ~ o ~+, where 0~ : C -- D ~ C -- D is an analytic function; by restriction to V+, ~ = ~ o ~+. But a look at the functional equation shows that e must be of the form ~ ~-~ a~ k, with k a positive integer and a d- ~ = 1. Now the asymptotic expansion shows that a ----= 1 and k = 1. [] 9. The canonical compactificatlon of K+ Let C be the compactification of C adding a circle at infinity. Then any poly- nomial extends continuously to C and its restriction to the circle at infinity multiplies angles by the degree of the polynomial. This section contains a description of an analogous compactification C ~ of C 2, to which H~non mappings extend continuously. A 3-sphere is added at infinity and the mapping extends as the solenoidal mapping a on S 8. This further emphasizes the similar role which solenoidal mappings play for Hdnon mappings and angle doubling plays for quadratic polynomials. In particular, the closures of K_ and K+ in ~2 are the solenoids 2]+ and Z_ res- pectively (note the reversal). This sometimes allows us to measure " angles of external rays " in K_ in the solenoid Y,+. This will turn out to be important in the description of the topology of these sets. Theorem 9.1. -- There exists a compact Hausdorff space X homeomorphic to a closed four-baU with underlying set C a  S s such that a) the induced topologies on C a and S 3 are the standard topologies; b) C a is dense in X; c) the H~non mapping F extends to a homeomorpkism F : X ~ X; and d) the restriction of F to S 3 is the solenoidal mapping v o. HI'NON MAPPINGS IN THE COMPLEX DOMAIN, I Remark. -- This is a surprisingly difficult result to prove, considering that the analogous results have already been worked out at finite distance. The difficulty is that if a 3-sphere is added at infinity in the obvious way, with points corresponding to oriented directions in C 2, then all non-vertical directions are mapped to horizontal directions. In particular, the Hdnon mapping does not extend continuously, and even where it is defined, it fails to be injective. To make the extension of the Hrnon mapping injective, the horizontal (and vertical) directions will be examined with a microscope. More precisely, a delicate blowup of the circle at infinity in the x-axis will be made, replacing a point p by a way of approaching p, the method of approach which we focus on being the images of straight lines. Proof. -- Step 1. Blowing up a circle in S 3. Consider a compact differentiable curve F C S ~, and define the oriented blowup g~ of S 8 along r as follows. First choose a tubular neighborhood U of F such that there exists a unique geodesic of S ~ joining any point of U to F in U. For x ~ U, let ~ be the tangent vector at x to the geodesic joining x to F. Now define the blowup Ur as a subset of the unit tangent bundle TI(U ) to U C S 3 to be ~jr={(x,~) ~Tl(SS) l{~:k~f~176 is perpendicular to I ~ if x ~ r The obvious mapping ~ : Ur -+ U is an isomorphism on Ur -- ~-~(r), so glue S 8 -- F onto Ur to make ~. Above F, there is a torus, mapping to F as a bundle of circles. Step 2. A first microscope. Consider the solid torus T 1 parametrized by {(X,~) ~CZllX[ = 1,]~[~< 2}. The point (X, ~t) of this solid torus will be " at the end of " the ray t~ t~tj. Let F 1 be the curve of equation y = 0 in S 3. Glue 0T1 = { (~, ~) ] I X [ : 1, I ~ I = 2 } to 2"8 as follows: choose the circle in Ts above the point of F1 corresponding to ~- 1" 1 ~1"1 the point in the circle at infinity on the x-axis in the direction X. Radial projection of the curve t~..[ xt'] L r~tJ onto the sphere at infinity gives a curve which approaches the circle orthogonally in a definite direction. Identify (X, ~) with this direction. Let $3 be the 3-sphere blown up along r 1 with the torus T 1 attached as above. 6 42 JOHN H. HUBBARD AND RALPH W. OBERSTE-VOKTH Step 3. A second m~croscope. Unfortunately, distinguishing these " eventually horizontal curves " do not resolve the images of straight lines. Blow up the circle U S of equation Vt 2 = X in this torus further to see the constant term in a ray of the form More formally, consider the solid torus T2 I = 1, 14 la]). Glue T, to S~, by identifying the point (~, v) e 0T~ (i.e. I v I = [ a I) to the unit vector in the direction of v at (~, ~). Let S~ be S~ blown up along I' 2 with the torus T2 attached as above. Step 4. A topology on C 2 U ~s. A basis of neighborhoods of each point ofS s is needed. There is no difficulty at those points which correspond to points of S 3 not on the x-axis: take the cone over a neighborhood of such a point, and cut it off at some radius. It is not much harder to define a basis of neighborhoods of a point (X0, ~0) in Tt which is not on the circle F2 or on the boundary of T 1. Take a neighborhood v--((x, )llx-z01< 01< of (X, bt) e T1 and let the neighborhood consist of V and the points which can be written (Xt ~, Bt) for t > T and (X, ~) e V. An analogous description is possible for the points inside T 2. Given ~0 with ] t~o ] = 1 and v 0 with ] v o [ < [ a [, choose a neighborhood W of (b~0, v0) in the solid torus T2 defined by ]mu--tZo]<r o[<,, and a number T>0. Then a neighborhood of (~to, v0) will consist of W and the points of C 2 which can be written (~t 2 t 2, ~tt + v) with (~, v) E W and t> T. It is a good bit harder to define a neighborhood basis for a point on the boun- dary OT 1 or 0T 2. Let P be the solid paraboloid of points in C ~ which can be written (Xt 2,vtt) with [~t]< 2 (i.e. the set defined by the inequality ]y]2~< 4]xD. Choose (X0, ~t0) e 0Ta, i.e. ] X 0 I = 1 and [ ~0 ] = 2, and a neighborhood W1 of (X0, Vt0) in T 1. The intersection W 1 c~ 0T 1 corresponds to a set of unit vectors normal to r 1. Set W z to be the set of points in S 3 which are obtained by traveling a distance less than ~ from I~t on the geodesic tangent to such a normal vector. Now a neighborhood of (X0, ~t0) consists of Wt u W~, and the points which can be written (Xt ~, #) with (X, ~) e W 1 and t > T, and the points on rays through W2 of norm greater than T and outside the paraboloid P. We will leave to the reader the analogous construction for the boundary 0T2, as well as the proof of the following lemma. Lemma 9.2. -- The space C 9" U ~s with the topology above is compact Hausdorff. Step 5. The space C ~. The compacfification of C 2 constructed so far is adequate to do the HEnon mapping once in appropriate regions. It needs to be adapted to the HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I 43 H6non mapping as a dynamical system. One way of doing this would consist of making an infinite sequence of " blow ups" as above, so as to resolve the images of parabolic rays, etc., and taking the projective limit. A different method will be used, inspired by the fact that a model for the locus at infinity already exists, as a dynamical system. Let S~ be a " new " copy of the 3-sphere, and ~ : S~ ~ S~ a solenoidal mapping. Let T' 0 C $1 s be a solid torus, such that T~ = a(T'o) is contained in the interior of To. The reason for this funny notation will become clear below. Back in 2 3, consider the solid torus T O of equation 15 [ ~< I x [. A ray in the cone over the boundary of T O can be written (t% t~) with [ ~ [ = [ ~ [ = 1. The image of such a ray is the parametrized curve t ~ (~a tz -t- c -- apt, ~t), which is asymptotic to the curve (9.3) s ~-~ ~a s z, ~s -t- ~ 9 This last curve converges as s ~ oo to a point in 0T2, and it is easy to see that the first is sufficiently close to the second so it converges to the same point. o o I -- P Choose a homeomorphism h:T 0 T 2 ~T o T2, conjugating the mapping : 0T 0 -+ 0T~ to the mapping T O ~ T 2 induced by the Htnon mapping as above. This is possible, by the classification of solenoidal mappings and the formula (9.3). Now put a topology on C ~ U S, 8 as follows. Attach T o --T' 2 to C 2 by k. For any point p of S~ -- (~+ u Z_), choose n such that ~(p) e T o -- T;, choose a neighborhood U of ~"(p) and define a neighborhood ofp to be F ~ c~ C 2) u ~~ n S~). This defines a neighborhood basis for all points in C 2 t_A SSl, except for those of the solenoids l~+ and Y~_. Recall that both solenoids are canonically homeomorphic to lira (S x, 2), and that there exists a unique mapping t~+ : T O ~ S 1 which semi-conjugates to angle-doubling (Theorem 3.1). A neighborhood of p = (..., Pz, Pl, P0) e Y~+ is the union of the set of (x,y) ~ C a such that F~ for all k~<N and ] v+(FO_k(x,y))[ --Pk < r for all k ~< N and the set of points p in n o -~ To k~N such that I t~+(a~ --Pk ] < r The union C a II $81, with this topology, is the space C 2. Step 6. Compactness of C a. It remains to verify that C a is compact, Hausdorff, and that the mapping F : C 2 ~ C 2 which is F on C 2 and ~ on S~ is continuous. 44 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH To show compactness, take a sequence in C 2. Suppose that the sequence is in C ~ and that the norms of the elements (x.,y.) tend to infinity, as otherwise the sequence obviously has a convergent subsequence. Further, assume that the sequence lies in V+. Either there exists k and a subsequence (x.l,y.~) such that F~ r V+, or there isn't. In the first case, recall the set P from step 4, and choose k and a sub- sequence, which will still be called (x,l,y,i) such that FO-ktx ~,,)'nl/ . ~ V+ mp. If a subsequence can be chosen so that the rays through these points converge to a non- horizontal ray, then this subsequence converges to the point of S~ corresponding to this ray. If the rays through these points tend to the horizontal, then choose a subsequence so that the directions of the rays tend to I~1 on a curve orthogonal to I" 1. This direction is then a point of S~, which is the limit of the subsequence. This shows that the F ~ ~(x m,y./) have a subsequenee which converges, and hence so does (x,i,y,~), by the second part of sept 5. Now suppose that the number of times F-1 can be iterated on points of the sequence and stay in V+ tends to infinity. Then by the compactness of the circle and a diagonal argument, a subsequence of the (x,,y,) can be chosen so that the sequences ] 9+(V~ [ converge for all k, say to Pk. Clearly p~ = P~-I. Let p = (..., p~, Px, P0). The subse- quence clearly converges to p ~ Y~+. This shows compactness. Step 7. The space C 2 is I-Iausdor~. It remains to show that distinct points of C~ have disjoint neighborhoods. Clearly only points in S~ need to be considered, and Lemma 9.2 shows that only points in the solenoids need to be considered. Even here there is no problem. If p0 ~: P'o, then let s = ~ [Po -- Po ]- Then the s-neighborhoods of p and p' are disjoint. If p, = p~ for i < k and Pk ~e p~, then [ Pk -- P~ [ = 2 and so the ~-neighborhoods are disjoint. Since clearly the mapping if, given by F on C * and a on S s, is continuous (the bases of neighborhoods are invariant under F), this ends the proof of Theorem 9.1. [] Corollary 9.4. -- The closure of K+ in C 2 is K+ u ~_ , and the closure of K_ is K_ u ]g + . Proof. -- Clearly points of K and large norm are points where ~- 1 can be iterated many times staying in V+. Such points are in smaller and smaller neighborhoods of points of Y,+. [] 45 HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I REFERENCES [A] AHLFORS, L., Conformal Invariants, McGraw-Hill, New York, 1973. [B] BEDFORD, E., Iteration of polynomial automorphisms of C 2, Proceedings of the International Congress of Mathe- maticians, 1990, Kyoto, Japan, Springer-Verlag, Tokyo, Japan, 1991, 847-858. [BSl] BEDFORD, E. and SMILUE,J., Polynomial diffeomorphisms of C~: currents, equilibrium measure, and hyper- bolicity, Invent. Math., 103 (1991), 69-99. [BS2] BEDFORD, E. and SmLLm, J., Fatou-Bieberbach domains arising from polynomial automorphisms, Ind/ana U. Math. J., 40 (1991), 789-792. [BS3] BEDFORD, E. and SmLLm, J., Polynomial diffeomorphisms of C ~, II: Stable manifolds and recurrence, J. Amer. Math. Soe., 4 (1991), 657-679. [BS4] BEDFORD, E. and SMILLm, J., Polynomial diffeomorphisms of C ~, III: Ergodicity, exponents, and entropy of the equilibrium measure, Math. Ann. (to appear). [BLS] BEDFORD, E., LYUBICH, M. and SMILLIE, J., Polynomial diffeomorphlsms of C 2, IV: The measure of maximal entropy and laminar currents, (to appear). [BC] BENEDICKS, M. and CARLESO~, L., The dynamics of the Hdnon map, Ann. Math., 138 (1991), 73-169. BIEBERBACH, L., Beispiel zweier ganzer Funktionen zweier komplexer Variabeln, welche eine schlicht [Bi] volumetreue Abbildung des R 4 auf elnen Tell seiner selbst vermitteln, Sitzungsber. Preuss. Akad. Wiss. Berlin, Phys.-math. Kl. (1933), 476-479. [BH] BRANNER, B., HUBBARD, J., The dynamics of cubic polynomials, I I: Patterns and parapatterns, Acta Math., 169 (1992), 229-325. [Br] BI~OL*N, H., Invariant sets under iteration of rational functions, Ark. Math., 6 (1965), 103-144. [D] DOLD, A., Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology, sk (1965), 1-8. [Dn] DOUADY, A. et HUBBARD, J., Etude dynamique des polyn6mes complexes, Publications math4matiques d'Orsay, Universlt4 de Paris-Sud (1984-1985). EARLE, C. and EELLS, J., A fibre bundle description of Teichmfiller theory, J. Diff. Geom., 3 (1969), 33-41. [EEl [F] FATOU, P., Sur les fonctions m4romorphes de deux variables, C. R. Aead. Sc. Paris, 175 (1922), 862-865; Sur certaines fonctions uniformes de deux variables, ibid., 175 (1922), 1030-1033. [FS] FORN,~SS, J. and SXBONY, N., Complex Hdnon mappings in C 2 and Fatou-Bieberbach domains, Duke Math. J., 65 (1992), 345-380. [FM] FRIEDLAND, S. and MILNOR, J., Dynamical properties of plane polynomial automorphisms, Ergod Th. & Dynam. Syst., 9 (1989), 67-99. GUNNINO, K., Introduction to Holomorphic Functions of Several Variables, Vol. III: Homological Theory, Wads- [O] worth & Brooks]Cole, Belmont, CA, 1990. HAGSTROM, M., Homotopy groups of the space of homeomorphisms, Ill. J. Math., 10 (1966), 563-573. [Ha] [H41] H~NoN, M., Numerical study of quadratic area preserving mappings, Q. Appl. Math., 2"1 (1969), 291-312. [H42] H~sos, M., A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77. HOLMES, P., Bifurcation sequences in horseshoe maps: infinitely many routes to chaos, Phys. Lett. ,4, 104 [Ho] (1984), 299-302. [HWh] HOLMES, P. and WmTLEY, R., Bifurcations of one- and two-dlmensional maps, Philos. Trans. Roy. Soc. London, Set. A, 311 (1984), 43-102. HOLMES, P. and WXLLXA~S, R., Knotted periodic orbits in suspensions of Smale's horseshoe: torns knots [HWi] and bifurcation sequences, Arch. Rational Mech. Anal., 90 (1985), 115-194. HUBBARD, J., The H4non mappings in the complex domain, in Chaotic Dynamics and Fractals (M. Barnsley [H] and S. Demko, ed.), Academic Press, New York, 1986, pp. 101-111. [HY] HUBBARD, J., Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoeeoz, in Topological Methods in Modem Mathematics: A Symposium in Honor of John Milnor's Sixtieth Birthday, Publish or Perish, Houston, Texas, 1993, pp. 467-511. HUBB~mD, J. and OBERSTE-Vo~TI~, R., H4non mappings in the complex domain, II: projective and [Ho] inductive limits of polynomials, in Real and Complex Dynamics, Kluwer, Amsterdam, 1994. MILNOR, J., Non-expansive H~non maps, Adv. in Math., 69 (1988), 109-114. [Ml] 46 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH [M2] MILNOR, J., Dynamics in one complex variable: introductory lectures, preprint, Institute for Mathematical Sciences, strNy, Stony Brook (1990). [MV] Mogh, L. and VXANA, M., Abundance of strange attractors, Acta Math. (to appear). [O] OB~gSTE-VoRTH, R., Complex horseshoes (to appear). [Sin] S~ALE, S., Differentiable dynamical systems, Bull. Amer. Math. Sot., 78 (1967), 747-817; reprinted in The Mathematics of Time, Springer-Verlag, New York, 1980. [S] SMmLIE, J., The entropy of polynomial diffeomorphisms of C ~, Ergod. Th. and Dynam. Syst., 10 (1990), 823-827. [T] THtrRSTON, W., The combinatorics of rational maps (to appear). [vD] VAN D~a~TZIO, D., l~ber topologisch homogene Kontinua, Fund. Math., 14 (1930), 102-105. IV] VIETORIS, L., l~lber den h6heren Zusammenhang kompakter Raume und eine Klasse von zusammen- hangstreuen Abbildungen, Math. Ann., 97 (1927), 454-472. [W] WILLn~S, R., One-dimensional nonwandering sets, Topology, 6 (1967), 473-487. [Y] Yoccoz, J., Sur la connectivit~ locale de M, unpublished (1989). J~ H~ Cornell University Ithaca New York 14850 R. O.-V. University of South Florida Tampa Florida 33620-5700 Manuscrit re~u le 5 janvier 1993 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Hénon mappings in the complex domain I: The global topology of dynamical space

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Abstract

HI'NON MAPPINGS IN THE COMPLEX DOMAIN I: THE GLOBAL TOPOLOGY OF DYNAMICAL SPACE by JOHN H. HUBBARD and RAI,VI-I W. OBERSTE-VORTH CONTENTS 1. Introduction ............................................................................. 5 2. An algebraic characterization of H6non mappings ............................................. 9 3. Solenoidal mappings ...................................................................... 11 4. Embeddings of the solenoid in S 8 .......................................................... 18 5. The functions G~ and $& ................................................................ 24 6. The global topology of H~non mappings ................................................... 28 7. The foliations of U+ ..................................................................... 32 8. An analytic description of U+ ............................................................. 35 9. The canonical compactification of K+ ...................................................... 40 I~FE~NCES ................................................................................. 45 1. Introduction In 1969, H6non ([H6I] and [H62]) began the investigation of the mappings F: ~ , where a+0, as mappings having roughly the same behavior as a particular Poincar~ section of the Lorenz differential equation. H~non demonstrated numerically that for certain values of the parameters the mappings appeared to have a strange attractor. This has finally been established rigorously by Benedicks and Carleson ([BC], [MV]). There has since been an enormous amount of work on the dynamics of the H~non mappings (in particular, see [Ho], [HWh] and [HWi], which give further references). This work is all in the real domain. As far as we know, this paper ([H] was an early version) is the first attempt to understand the H~non mappings in C ~'. Recently others have done work in this area including Friedland and Milnor ([FM] and [M1]), Bedford, Lyubich, and Smillie ([B], [BS1], [BS2], [BS3], [BS4], [BLS], IS]), and Formess and Sibony ([FS]). In the study of iteration of polynomials of one variable, extending to complex values of the variable has been very useful, even when the original polynomials were JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH real. We hope that the same thing will happen here, more or less for the same reason. There is essentially nothing that can be said about real polynomials which is independent of the coefficients, largely because virtually all features independent of conjugation, such as periodic cycles, are likely to disappear under perturbation. In the complex domain, the behavior is far more uniform. Our work started from a different point of view. In 1982, Calabi suggested that the computer should be used to investigate the basin of attraction of one of the two attractive fixed points of the mapping (11 ) (;) The reason for examining this was that it provided an example of a Fatou-Bieber- bach domain. These are open subsets U C C" which are biholomorphically isomorphic to C" and whose complement, C ~ -- U, has non-empty interior. When an automorphism of C" has an attractive fixed point or attractive cycle, the basin is always such a domain. Fatou and Bieberbach ([17], [Bi]) first constructed examples of such domains as basins of attractive fixed points. They have been extensively studied in [BS2] and [FS]. 9 Despite considerable numerical work, we were unable to work out the topology of the closures of the basins and decided to look at simpler automorphisms of C ~, with quadratic polynomials as coordinates. Section 2 shows that the Htnon family encompasses a significant part of this family. The east of players. Most of the work on Htnon mappings in the real case has focused on attractors. In the complex, attractors are uninteresting since the only attrac- tors are points. The invariant subsets considered here are inspired by the dynamics of polynomials, as explained below. For any mapping f, letf ~ denote the n-fold composition off or f-1 depending on whether n is positive or negative. Our approach has been inspired by the study of complex polynomials of a single variable. Given a polynomial p(z), the natural set to study is K~ = { z [p~ does not tend to oo as n ~ ~ } and its boundary J~ = 0Kr, also known as the Julia set ofp. Another definition of J~ is J~ = { z [ on no neighborhood of z is the sequence { pO, } normal }. The sets studied here are defined in imitation of the one-dimensional case. For a Htnon mapping, the obvious generalization of the Julia set is J~ ={(;)~ n~ neighb~176176 ~ (;) is the sequence { F~  } n~ } ' HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I where a sequence of functions on U C (I 3 with values in C ~' is defined to be normal if every subsequence has a subsequence which converges uniformly on compact subsets to a function with values in p2, the complex projective plane. Define for a Hdnon mapping F the following sets: = C 2 -- K+ [l o (;Ill oo-o,,en o } and -~C2--K . Further, define J = OK:~, K ----- K+ c~ K_, and J = J+ n J_. It will be seen that K and J are compact and of course invariant under F. These are the spaces which we most wish to understand. Ma~n results. This paper contains three main results: two concern the structure of C 2 -- K~. Topologically, we will show that this set is homeomorphic to a fibration over the reals with fiber a 3-sphere with a solenoid removed (Theorem 6.1). Analytically, C 2 -- K is isomorphic to a quotient of (C -- D) X C, where D C C is the unit disc, by a group of automorphisms which we determine explicitly (Section 8). The third result gives a compactification (I 2 to which the Hrnon mappings extend canonically, analogous to compactifying C by adding a circle at infinity (Theorem 9.1). The proofs of these results require both some analydc and some topological pre- liminaries. Most of the topology (Sections 3 and 4) concerns solenoidal mappings, one of which plays much the same role with respect to Hrnon mappings as multiplying angles by d does for iteration of polynomials. We go into more details than is strictly necessary for our purposes, but we feel that viewing the surrounding countryside makes our pard- cular mappings easier to understand, and the classification of solenoidal mappings (Theorem 3.10) is of independent interest. For the analytical results, the most important construction is the analog of the Brttcher coordinate ([M2]). When p is a monic polynomial, this is the function ?~ defined in a neighborhood of o0 such that and ~(z) = z + 0(1) near oo. The function q~(z) is constructed by making sense of the following O~(z) = lim (p~ n ..-~ oo This is a standard scattering theory construction: go toward oo via p and return via the unperturbed mapping z ~ z d. The fractional power is not a priori defined, and has to be dealt with carefully. JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH The problem of the branches of the roots can be circumvented by defining G~(z) = lirno ~; log+ Ip~ where log+(x) = sup{log(x), 0 }, which is the Green's function of K~. This construction generalizes for H6non mappings as follows. Let a subscript I or 2 denote the projection onto the first or second coordinate as in (F~ = pr t o F ~ Now define the limits (;) 1 G -= lim ~ log+ II F~  "(x) II, of course, the matter of where these are defined and the convergence of the limits must be dealt with (and are, in Section 5). Since the first version of this paper was written, much further work on G has been done, more particularly by considering the closed (1, 1)-currents ~  = dd ~ G+ which are analogs of the Brohlin measure ([BS1], [FS]). The measure ~ = ~+ ^ ~_ has also turned out to be very important. As far as we know, the complex analytic mappings ? have not received similar attention, but they are even more important to our development. More particularly, the argument of the B6ttcher coordinate has led to the theory of external angles and is fundamental to the combinatorial study of the dynamics of polynomials ([DH], [T]). When the functions q0 are combined with the compactifica- tion in Section 9, more particularly Corollary 9.4, we find that there is an analogous theory of external angles for H6non mappings; perhaps we can hope to use the techniques using external rays, etc., to combinatorially describe H6non mappings. A case in point is the Benedicks-Carleson result in [BC], where the combinatorics is so reminicent of puzzles and tableaux as in [Y], [BH] and [HY]. Continuations of this paper will present results about Hdnon mappings as pertur- bations of polynomials ([HO]). The paper [O] studies the dynamics of complex horse- shoes using techniques from these papers. Acknowledgments. -- This paper has taken a long time to write and we have benefited from innumerably many conversations. We particularly want to thank A. Douady, J. Smillie, E. Bedford, B. Branner, J. Milnor, D. Sullivan, C. McMullen, J. C. Yoccoz, M. Shishikura, N. Sibony, L. Carleson, M. Benedicks, C. Bardos, T. Bousch, D. Faught, and L. Ma. Much computer experimentation went into the formulation of the theorems, and we thank H. Smith for the immense amount of programming he performed for us over the years. HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I No scientific research is possible without financial support: we thank the National Science Foundation for grant DMS 8901729 and earlier grants, the mathematics depart- ments of Cornell University, Yale University, the University of South Florida, SUNY at Stony Brook, the Universit6 de Paris-Sud, the Institute for Advanced Study, the Geometry Project at the University of Minnesota, the Max Planck Institut i'tir Mathe- matik, the Institut des Hautes Etudes Scientifiques, and the Ecole Normale Supdrieure of Paris and Lyon for support and hospitality during the years when this paper was conceived and written. 2. An algebraic characterization of H~non mappings The family of mappings on C a with quadratic coordinate functions depends a priori on 19 parameters. The H6non mappings = , a4= 0, represent some conjugacy classes of quadratic automorphisms. In this section it is shown that the only other conjugacy classes are represented by the elementary mappings, where k 1 = 1 implies b = 1 and k 2 = 1 implies either b = d+ 1 or d---- 1. Note that the elementary mappings consist of several one- and two-parameter families: A +&t' b,0, b, 1, D = Note that every polynomial mapping, G, of degree 2 can be written in the form 13 =- G0 + G1 + G~, with each G k homogeneous of degree k and that every polynomial automorphism has constant Jacobian determinant. The following theorem gives the Jacob!an Conjecture in this context, i.e., any polynomial mapping of degree 2 with non-zero constant Jacobian determinant is an automorphism. Remark. -- If G~ satisfies the non-degeneracy condition G~-I(0) ----- 0, then the mapping G extends to give an endomorphism of P~, which will be of degree 4. More JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH generally, if the mapping were given by polynomials of any degree d, and the leading terms G a were non-degenerate, then the mapping defines an endomorphism of pa of degree d a. Of course, this is incompatible with G being an automorphism. Since G a is degenerate, G~-l(0) is a line t0, and Ga(C ~) is a line r Either go and gi coincide, in which case the mapping is elementary, or they do not coincide, and the mapping is a genera- lized Hdnon mapping. Theorem 2.1. -- For every polynomial mapping G : C a -+ C 2 of degree ~ with constant non-zero Jacobian determinant, the image of Gz and the set on which Gz vanishes are lines through the origin. If these lines are linearly independent, then G is conjugate to a Hdnon mapping. Otherwise, G is conjugate to an elementary mapping. Proof. -- The general polynomial mapping of degree two is = as+bax+cay+dax 2+eaxy+fayz]" The quadratic terms of the Jacobian determinant generally yield the relations dl ei fi eg fa ' da So the image of the quadratic terms is a line, gl; assume that dz = e9 =fa = O. The linear terms of the Jacobian determinant generally yield the relations e~ ba ca = 4dlf~ be ca. Since G is injective, bz and c a cannot both be 0. So Ga vanishes on a line, t o. Ifg o and ti are linearly independent, then assume di = 1 and el =fi = 0 (sending to to they-axis). The Jacobian condition shows that ca = 0 and this is a Hdnon mapping. Otherwise assume that fa = 1 and di = ei = 0 by sending to to the x-axis. The Jacobian condition shows that b 2 = 0. So G is of the form Note that bl 4= 0 and c a 4:0 are invariant under conjugations which do not introduce new terms. The different cases are listed below: ci(1 -- ca) = 0 yields B with b = d = b i = c2, c~4 = 1, b i=ca, 2aa+ ci(1 -- c~) 4= 0 yields A with b = b i ---- c~, c9 4 = 1, b l = ca, 2az + yields B with b ---- ba, d = ca, ca 4: 1, bi4:cz, b i4:1 ca4: 1, hi= I, al(l-- ca) a + aa(1 -- ca) + a~ = 0 yields B with b= 1, d=ca, HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I 11 cz4: 1, b 1= 1, ax(1--e2) 2+az(1-c2) +a~4=0yields C with d = cs, cs= 1, a s4=0yieldsDwithb=bl, c s= 1, a s-~O, b a4= lyieldsBwithb=bx, d= 1, cs= 1, a s=O, b 1= 1, 4ax--c~=OyieldsBwithb=d= 1, cs= 1, a s=O, bl= 1, 4a t-c~4=OyieldsCwithd~- 1. [] Remarks. -- The H6non family of mappings can be written in different forms. For example, Hdnon ([H~I] and [H62]) actually studied the family = ~x , ~ 4= 0. Note that Fa., is conjugate to H_,. _a. Thus, mappings of the form Fa, 0 were omitted in this other form. A fixed line is a line which is mapped onto itself (but not necessarily pointwise). Consider the set of lines y = k for all k ~ C. Elementary mappings can be understood by how they map these lines: all lines fixed, a unique fixed line, or no fixed line. A fixed line may be fixed pointwise, or there may be a unique fixed point or no fixed point. 3. Solenoidal mappings This section gives a classification up to conjugacy of unbraided solenoidal mappings, v:T ~ T of degree d, satisfying appropriate expansion properties and topological conditions. Solenoidal mappings, which are defined below, are injective mappings of degree d/> 2 of the solid torus. The images of such mappings can be braided and quite complicated. We only understand how to classify those which are unbraided. We will show that up to conjugacy, such mappings, when they are appropriately expanding and contracting, are classified by an integer. Only one of these mappings seems relevant to the study of H6non mappings. On the other hand, the authors puzzled about these mappings quite a bit while understanding the structure of H6non mappings, and we feel that it will be clearer if we study them all, if only to contrast the relevant one to the others. Theorem 3.1 holds for arbitrary mappings of degree d while Propositions 3.3, 3.5, and 3.6 require the mappings to be solenoidal. The construction of solenoids is given before Proposition 3.6. Proposition 3.7 shows that solenoidal mappings of degree 2 are unbraided while Proposition 3.8 requires unbraidedness. Theorem 3.11 is the classi- fication of conjugacy classes. Theorem 3.11 reduces the determination of a conjugacy class to the computation of an isotopy class and the verification of a hyperbolieity condition. 12 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Solenoidal mappings. Let D be the disk of radius 2, T = S 1 � D, and denote by (E, z) the coordinates in T. Definition. -- Let C+ and C_ denote the constant families of cones C+(~, z) ={(~,u I l~l~> lu[} and C_(E, z) ={(g,.) I Igl~< I"1} in the tangent bundle of T. Definition. -- A solenoidal mapping 9 : T -+ T of degree d is an injective C 1 immer- sion of degree d, such that, for all (E, z) e T and for some constant K > I, d,~,,, ~(c+(E, z)) c c+(<E, z)), (g, u) ~ C+(~, z) and dc~,, , -r(~, u) = (gx, ux) imply [gx [ > K IT [ and (g, u) e C_(E, z) and d,;,,, ~(~, u) = (gx, ux) imply [ u x [ < K [ u [. Remark. -- The definition says roughly that the derivatives of a solenoidal mapping preserve the family of cones G+ and are expanding in the g direction and contracting in the u direction in C+. From the fact that 9 is an immersion it follows that the inverses of the derivatives of a solenoidal mapping preserve the family of cones C_. Examples. -- Let S 1 --{E Cl I E I = 1 ),D ----{ zECl[ z ]~< 2},andT =S 1 � D. Define e 1 = { 1 } � 0 D and e~ = $1� { 2 ), each oriented by the counterclockwise orientation of the circle. We will examine very closely the following mappings, ~,,: T -+ T, which are unbraided solenoidal for every integer d t> 2 and k ~ Z: E d The reason for the shift in the exponent will become clear later: "~d, 0 has much nicer properties than the others. Theorem 3.1. -- For every mapping f: T -+ T of degree d >>. 2, there exists exactly d -- 1 continuous functions ~ : T ~ R/Z of degree 1 such that the following diagram commutes: T >T R/Z > R/Z For any two such mappings ~1 and ~z, there exists o with co a- 1 = 1 and ~x :- o~2. Proof. -- By the Lefschetz Fixed Point Theorem ([D]), f has 1 -- d fixed points counted with multiplicity. Since 1 --d 4= 0, there is at least one fixed point t 0. To avoid difficulties with branches of d-th roots, lift f to f~: T ~ 1", where HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I 13 = R � D is the universal covering space of T, with base point t o and the liftff: T -+ T satisfying 3~0) = ~0. Let y: ~"--> 'F be a generator of the fundamental group. Then a~V(7))---= y~ for every 7 e ~. Consider the space II = { ~ : "~ -+ R [ ~ is continuous and ~(y(t)) = ~(t) + 1 } with the uniform metric, 8, which is well-defined because of the periodicity. In this metric II is a complete metric space. Define a mapping q~ : II ~ 11 by ,(~) (7) = ~(f(7)). If ~ satisfies ~(ff(~')) = ~(~') + 1, then so does ~(~). Lemma 3.9.. -- The mapping ~ is strongly contracting. Proof. -- If ~1, ~ e II, then ... 1 1 ~ ~ I ~(~1) (7) -- ~(~) (7) 1 = 2 ] ~1(Y(7)) -- ~(f(7)) [ ~ d 8(=1, ~). [] (Lemma 3.2) Let ~o be the fixed point of ~ and n o be the mapping T ---> R/Z induced by ~o. Clearly the mappings rc k = e z'a~m-l~ % still semi-conjugate f to z ~-* z a. If rd : T -+ R/Z is any mapping making the diagram T >T ze-> z d R/z , R/z commute, then ='(to) is a fixed point of z F-+ z ~, so it must be one of the (d -- 1)-th roots of 1 and there exists k with ='(t0) --= =k(t0). Now r~' --- %, since the lift of e -2~/~a-~ re' in H is a fixed point of % hence is 7:o. [] (Theorem 3.1) Proposition 3.3. -- If a mapping f of degree d is solenoidal, then the mappings = : T -+ R/Z in Theorem 3.1 are fibrations with fibers homeomorphic to disks. Proof. -- Define H 0 C H, the family of IApshitz fibrations consisting of those ele- ments of H whose fibers are disks which are graphs of Lipshitz functions cr : D --> R (i.e. ] ~(zl) -- ~(z~) I ~< [zz -- z~ I)" Sinceff -x preserves the family of cones e_, the family of IApshitz fibrations is stable under ~. So the fixed point =o is a limit of IApshitz fibrations. The space II 0 is not closed in II, but the fibers of= e II 0 are fairly easy to understand. Lemma 3.4. -- For any rc = lim rc~, with rcj e IIo, and for any x eR, there exist two Lipshitzfunctions o~(z) <. %(z) such that =-~(x) = {(y, z) [ ~(z) ~<y ~< ~,(z)}. I4 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proof. -- We have = n u n , ;l(txo - xo + 9 >0 i>o j~>i Decreasing intersections of sets of the form r~-l(x) = {(y, z) I ~l(z) ~< y ~< 0c2(z)} are still of this form. Hence it is enough to show that U n ;l([xo - Xo + is of this form. Now suppose that r~-~([x0 -- ~, x o + e]) ={(y, z) ] %.~(z) ~<y~< ~j.~(z)}. Then U,>o o - Xo + = {(y, z) [ lim_.soou p. ~, ~(z) ~< y ~< lim,_.~oinf ~, ,(z)}. We need to show that %,~(z) ~< ~,~(z) for all j and ~ sufficiently large. For any fixed z, the function Y~-*~o(Y- z) is surjective, so there exists ~o with ~o(Y'o, z) = x 0. Choose I so large that 1~ -- ~o t < e forj >t i. Then ~j(y%, z) e [x o -- e, Xo + ~]. So for all j >i i, %.,(z) ~<Yo~ ~,,(z). [] (Lemma 3.4) Now if a fiber is not a Lipshitz disk, then it has nonempty interior. Also the fibers are compact and their projections onto R have bounded length. If any two points (xx, za) and (x2, zz) satisfy I z2 -- z~ ] < x 2 -- x~ and (x~, z~) =j~x,, z~), then x' 2 -- x'l >i K(xz -- xa) for some K > 1. If a fiber is not a Lipshitz disk, then let (xx, zx) and (x~, %) be two points of the interior satisfying [z 2 --z x 1< x~ -- x 1. Let ~,f""', z~"')=ff~ z~). These are still in the same fiber and x~- ~"~ -- x~ "1 i> K"(x z -- xx) for some K > 1. This contradicts that the length of the projection onto R of the fiber is finite. [] (Proposition 3.3) Proposition 3.5. -- The components 0ff~ n r:-l(z) have diameters tending to 0 as k -+ oo for all z ~ S 1. Proof. -- Suppose that x,y ~fk(T) are in the same fiber and realize the maximal vertical diameter, d~, offk(T). Compare this with the maximal vertical diameter, dk_l, of f k- I(T). Let xx ----- f- 1 (x) and Yl = f- 1 (y) and let t be the straight line joining them. The sdenoid N d. Given a space X with a mapping f: X-+ X, consider the projective limit Xt=lim(X,f) ={(...,x~,xl, xo) lf(x,+l) -=x i for i= 0,1,2,...}. When the mapping f is clear, simply write X instead of Xt" This construction is some- times referred to as the inverse limit construction. A point of this projective limit is a point of x 0 e X along with a " history of the point " under the iteration off HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I 15 The mapping f induces f: ~ -+ X by f(. . ., x,, xl, Xo) = (. . .,f(x2),f(xi),f(Xo)) -= (..., Xl, Xo,f(Xo) ) which is always bijective as f-l(..., xl, x0) = (..., xl). Consider the projective limit of the mapping 8:Si---~ S i given by ~(~)= ~a. Define Z a ----ljm(S i, 8) and the bijection ~:Z a -+ Z a as in the introduction above. This construction was studied carefully by Williams ([W]). The solenoid was first studied by Vietoris ([V]) and van Danzig ([vD]). Let r 9 be one of the d -- 1 mappings guaranteed by Theorem 3.1. Proposition 3.6. -- Let Z I = ~,f~ The mapping X v-~ (..., rr176 rcl(f-l(x)), x1(x), =i(f(x))) is a homeomorphism h~ : Xf ~ X a. Proof. -- Let z = (..., zl, z0) be a point of Z, and define X,, k = {x el" [f-'(x) is defined and ~/f-'(x)) = z, for 0,< i,< k}. Then fok maps ,:Ti(zk) bijectively to X,,~. In particular, Xz, k is a component of fo~(-~) n ~TX(z0). Since these components have diameters tending to 0 as k ~ 0% and h (z) = fl Xz, , we see that h~. is bijective. It is continuous, and the domain is compact, so it is a homeo- morphism. [] Definition. -- An injective mapping v : T ~ T of degree d is unbraided if there exists a fiber homeomorphism q~ : T -+ T such that @ o -: sends the core circle S ~ X { 0 } into S 1 � S i as a (d, 1)-torus (un)knot. Fro. 3.1 16 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Remark. -- No embedding of T in S 3 is specified. In particular, the (d, 1)-torus (un)knot and (d, d + 1)-torus knot are equivalent from this point of view. Proposition 3.7 shows that ff d = 2, all solenoidal mappings are unbraided; this is false for d t> 3. See Figure 3.1. Proposition 3.7. -- For degree d = 2, all solenoidal mappings are unbraided. Proof. -- We will show that there exists a fiber homeomorphism g: (T,f(S z X {0})) -+ (T, %,k(S 1 � {0 })). Note that the pair of sets (T, %,k(S 1 � { 0 })) is independent of k. Both (T,f(S t � (0))) and (T,-r ~ � (0))) are locally trivial fiber bundles over S 1 with fibers homeomorphic to disks with two marked points. Locally trivial fiber bundles over S 1 are classified by the isotopy classes of their monodromy. For the bundles under consideration the monodromy homeomorphisms lie in Womeo (D, { a, b )). The mapping M'omeo (D, { a, b }) -> ~erm { a, b } (the latter being the symmetric group on two elements) is surjective with contractible fibers ([Ha]). So the isotopy class of the monodromy of these bundles depends only on how they permute the boundary components. For both bundles, the points are exchanged. So the bundles are fiber homeo- morphic. [] Proposition 3.8. -- For every unbraided solenoidal mapping f there exists a unique integer k and a mapping h:T- int(f(T)) -+T- int(-~d,,(T)) suck that the following diagram com- mutes: OT > OT (3.9) 0f(T) > ~d,,(T) Proof. -- Step 1. There exists a fiber homeomorphism g : T -- int(f(T)) -+ T -- int(va, k(T)) mapping 0T to ST. Note that the set T --int(vd,,(T)) is independent of k. The definition of unbraided says that the bundles of pairs (T,f(S 1 � { 0 })) and (T, vd,,(S 1 � { 0))) are fiber-homeomorphic. Pick a base-point in S 1, and let D be the fiber above that base-point, { at,..., aa} = D nf(S1 � {0 }), with the points ordered along the circle f(S 1 � {bl, ...,bd}=D c~%.k(S 1 X{0}) ordered similarly. Then the bundles of pairs above are classified by their monodromies m I and ma. k. The definition of unbraided says that there exists a homeomorphism q~ : D --> D with c?({ al, ..., aa }) = { bl, ..., b a } and conjugating m s to md. k. Let U1,..., Ud be the components of D nf(T), labeled so that a~ ~U~, and Vt, ..., V a be the components of D n va.k(T), labeled so that b~ ~V~. Now deform 9 HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I so that 9(U~) = V~. This is (unpleasant) 2-dimensional topology. First adjust ~ in the definition ofzd, ~ so that $(U~) c~V~ = 0 for issj. Therefore the sets Vi II $(V~) have disjoint neighborhoods D~ homeomorphic to disks. Lemma 3.10. -- If U and V are closed subsets of the open unit disk D, with 0 e (J c~ ~r and both homeomorphic to closed disks, then there exists a homeomorphism ~ : D ~ D which is the identity on 0 D and with ~b(U) = V. Proof. -- Use conformal mapping to represent both D -- U and D -- V as stan- dard annuli, giving a system of" polar coordinates " where the radial curves are labeled by the points at which they intersect 0 D and the circular curves by their relative distance to 0 D. Then making points with the same coordinates in D -- U and D -- V correspond gives a homeomorphism of D -- l~l onto V -- V. This can be continued to U and V since any homeomorphism of the boundary of a disk extends to a homeomorphism of the interior, by radial extension, for instance. [] (Lemma 3.10) Find a homeomorphism d~: (D,{ bx,..., ba})-+V such that + o q~ is isotopic to ~p and + o ~p(U~)= V~. Unfortunately, this mapping does not now conjugate the monodromies, but it does up to isotopy, and that is enough, since bundles are dassified by the isotopy classes of their monodromy. Step 2. Next it will be shown that k can be chosen so that diagram (3.9) commutes on the level of homology. The homology group HI(0T ) is isomorphic to Z 2. Choose the basis { S 1 � { 2 }, { 1 } x 2S x }, the circles oriented counterclockwise in C. Consider the mapping g of- x o g- 1 o T 0. This is a fiber homeomorphism 0T -+ ~I', hence induces a mapping given by a matrix (I 0 i) for some integer t on HI(0T). Since the construction of g is unique up to isotopy, t is an invariant off. Observe that ~a,, can be written -ra, 0 o w ~, where w is the twist mapping w(L = So g of- 1 o g- 1 o "ra, k = g of- 1 o g- 1 o "ra, o o w ~. Since w ~ induces the mapping given bythematrix (; kl) on Hl(OT), set k= --t so that gof-log-~o~d, k induces the identity on the homology. Step 8. Finally, adjust g into h so that the desired diagram commutes. There exists a homotopy G~:0T X I~T with Go=.r~ogofog -1 and G~ = Id. Let U C T- int('ra, k(T)) be a narrow thickening of 0T homeomorphic to I � OT. Denote points in U by (t, x). Let 'g(y), y Cg-a(U) h(y) = G,(x), y eg-a(U) so that g(y) = (t, x) 3 18 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH on T -- int(f(T)). Now for y ~ ST, set x = g(y) and compute (f-loh-1 va,~oh ) (y) =f-~og-io.ca,~o (z~,~ og 0 --1 ofo g-~) (x) : g-l(x) :y. [] (Proposition 3.8) Next comes the classification of the conjugacy classes of unbraided solenoidal mappings. Theorem ~.11. -- Every unbraided solenoidal mapping is conjugate to one of the va, k, and no two of these are conjugate. Proof. -- The second part was proved above, when it was shown that different values of k lead to different values of l, which are conjugacy invariants. We wish to extend h from Proposition 3.8 to T. Take x ~ T. If x E X:, then define h(x) = ~a:k o ~: ~ X~a,k and if x r X:, then define h(x) = Va,~ o g :, where m is such that f~ int(f(T)). If f ~ 0f(T) so that f~ 0T, then both choices, m and m + 1, give the same value of h by Proposition 3.8. So the mapping is well-defined, bijective, and conjugatesfto va, k" It remains to show that h is continuous on Z/. Take x 0 ~ X:. The sets U~, ~(x0) = { x ~ T ]fo-.(x) exists for all n ~< N and I ~f(fo- ~(x)) -- ~s(f ~ ) I <~ ~ for n ~< N } form a basis of closed neighborhoods of x 0 as r -+ 0 and N -+ ~. Clearly k maps this basis of neighborhoods of x 0 to the corresponding basis of neighborhoods of f(xo). So h is continuous and hence a homeomorphism. [] 4. Embeddings of the solenoid in S a In this section we will try to describe the inductive limit of T under -% 0. Intui- tively, this corresponds to taking a solid torus winding around d times in a larger torus which winds around d times in a yet larger torus, etc., and taking the increasing union. This intuitive picture is ambiguous. To make this precise, the embedding mapping each torus into the next must be specified. This is made precise in this section. Smale first studied solenoids as hyperbolic attractors in S 3 ([Sm]). Recall the mappings ~a,, from the example near the beginning of section 3. Proposition 4.1. -- The mappings %,o extend to orientation-preserving homeomorphisms h a : S a -~ S a. Remark. -- Note that *a,k obviously extends to S a for some k. After all, one can take a solid torus (think of a bicycle tire tube) and wrap it d times around itself. The outside of the unwound tube and of the wound tube are both unknotted tori, so there exists a homeomorphism between them. This homeomorphism will map the inner rim ~oa,~-,~tx~ HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I of the tube to some curve on the wound tube; the object of this proposition is to describe this curve. The skeptical reader might experiment with a tube for d = 2, 3. The proof of Proposition 4.1 depends on the following: Lemma 4.2. -- If T 1 and T~ are two solid tori, and f: OT 1 -+ OT~ is a homeomorphism which sends curves on OT 1 which bound disks in T 1 into curves which bound disks in T~, then f extends to a homeomorphism T a -+ T~. Proof of Lemma 4.2. -- We may suppose T 1 = T 2 = T = S 1 � D. The homeo- morphisms of a torus are classified up to isotopy by their action on 1-dimensional homo- logy. If a homeomorphism of 0T extends to T, then any isotopic homeomorphism extends also. Clearly the linear homeomorphisms mapping curves of the form { ~ } � 0 D extend. [] (Lemma 4.2) 1/2~0 FxG. 4.1 Proof of Proposition 4.1. -- The key point is that ~a,o maps curves on 0T which bound embedded disks in S 3 -- T into curves which bound embedded disks in S 3 -- "ra, 0(T). This can be seen in Figure 4.1. This is a drawing of S 1 x D, with the disks { 1 } x D, { i } X D, { -- 1 } � D, { -- i ) x D; the reader is expected to fill in the other slices. Within these disks are d subdisks. The case d = 3 is represented, and the triangle formed by these three subdisks rotates by 1/3 of a turn while going around S 1 once. Thus these subdisks represent -rd, k(T). JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTI-I The curves drawn on the outside of the disks represent a disk X in S 8 -- "rd, k(T). Verifying that this is indeed a disk is the essence of the proof. We leave to the reader to verify that X is a manifold with boundary OX C 0~d,~(T). To see that this manifold is simply connected, notice that it is clearly a deformation retract of the subset consisting of X n S 8 -- int(S 2 -- T), and the star above -- 1. This is a contractible set: d disks, each with a leash and all leashes connected at one point, as in Figure 4.2. Now going around S 1 once, the angle at which X touches a subdisk rotates by -- (d- 1)/d, so that altogether 0X is the curve e ~'~t ~ (e 2d'n', e~,et + ce~im-d)). The mapping ~d,0 maps e~ (a curve bounding in S 3 -- T) to this curve, so ~a,0 extends as required, by Lemma 4.2. [] (Proposition 4.1) Fxo. 4.2 Reflections with respect to a torus. A different way of understanding the exten- sion of ~a,0 to S s will be given requiring a definition of reflection with respect to a torus. The simplest context in which to describe such reflections is to write S =={(u,v) l lul 2+1 vl 2= 1}. Then S s = T' to T", where T'={(u,v)~S'llulal/V'2} and T"={(u,v)~S31lvl<~l/V~}. These are two unknotted solid tori, and dearly they are exchanged by the mapping HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I 21 To give an intuitive description, we will work in S ~ = R 3 t3 { dO }. So this mapping needs to be translated into a mapping R a t3( do )~ R 3 u{ dO }. Stereographic pro- jection from the point (0, i) maps S ~ to R 3 ~3 do according to the formula Re u/(1 -- Im v)) (~)~ Imu/(1--Imv) . Re v/(1 -Im v) This mapping takes the torus l u [ = e~~ Iv [ = e~~ to the parametrized torus in R a /cos 01/(v - sin (01)0 v'-'/sin 01/(3/2 --sin 0~)~, \cos 0~/(A/~ -- sin 0~) ] which just happens to be the torus of revolution obtained by rotating the circle of radius 1 centered at (We2, O) in the (x, z)-plane around the z-axis. Conjugated by this change of variables, the mapping p becomes ( z ) x ~+ (y-- 1) ~+z ~ x~ + y~ + z2- 1 . 2x Note that Pl commutes with reflection in the y-axis. Fro. 4.3 Construction of h a. Consider two unknotted solid tori T O and T 1 embedded in R a, linked with linking number d, as in Figure 4.3. Then R d : S 3 -+ S 8, the rotation by 7rid around the z-axis, is a homeomorphism of each onto the other. For any homeomorphism ~ : S 3 -+ S 8 set p~ = ~-1 o Pl o ~. JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proposition 4.3. -- There exists an orientation-preserving homeomorphism such that a) ~ maps T O to T'; b) ~ commutes with reflection in they-axis; c) the restriction of h a : p~ o R a to T o is a solenoidal mapping conjugate to ~Va, o; d) h a is conjugate to its inverse. Proof. -- Fiber both T O and T1 over the circle by the radial angles, as measured from they-axis, and similarly for T'. Choose first the restriction of the homeomorphism to To, so as to map the slice with a given radial angle of To to the corresponding disk of T' and so that ~ commutes with symmetry with respect to the y-axis. Next choose a curve 7 winding d times around T', symmetric with respect to the y-axis and such that the " radial angle " of T" is monotone along the curve, and a small tubular neighborhood S around it. Note that this radial angle of T" will increase by 2dn along ~. Fiber S by the radial angle, starting at the highest intersection on the y-axis. See Figure 4.4. Fio. 4.4 Next, define 0~ on T 1 by sending the slice at a given angle to the slice of S at d times that angle, still preserving the symmetry with respect to the y-axis. Extend the homeomorphism to S 3 so as to preserve the symmetry. With this choice of ~, a) and b) are clearly true. All the work was designed to satisfy c) and d): the restriction of ha simply multiplies radial angles by d in To, hence is expanding in that direction. By choosing the tubular neighborhood S of y sufficiently thin, ha can clearly be made contracting in the slices. Since y is unbraided in T", the sole- noidal mapping ha : To -+ To is conjugate to va, k for some k, which must be 0 since ha extends to S 3. The inverse of ha is Ralo p,, which is conjugate to p~o Ra 1. Conjugate the mapping by symmetry around the y-axis. This conjugates R a to R~ -1, and since the HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I reflection with respect to the y-axis commutes with p,, R o P0 is conjugated to its inverse. [] This shows that the mapping ha : S 3 -+ S 3 is a homeomorphism, which has two invariant solenoids Z+ C T O and Y,_ C T1, attracting and repelling, respectively; every point is attracted to ~+ under forward iteration of ha and is attracted to Z_ under itera- tion of h~ -1. haduetlve llmlts. Given a space X and a mapping f: X -> X, define the inductive limit lim(X,f) to be li_m(X,f) = X x N[,~, where the equivalence is generated by setting (x, m) ,~, (f(x), m + 1). The notion of inductive limit is pathological when f is not injective (the spaces created fail to be Hausdorff). We will use the notion only for injective mappings f, where it really is some sort of increasing union. Proposition 4.4. -- The inductive limit lira(T,-cd,0) is homeomorphic to S 8- Z a and r.a = N.. ~2,o(T). Proof. -- The mapping (x, m) F-. k~-'(x) induces a mapping lim(T, ~a, 0) -->Um ha re(T0). The mapping ha is conjugate to its inverse, and the conjugating mapping is a homeo- morphism of S ~ -- h~-re(T0) onto ha(T0). [] Corollary 4.5. -- The fundamental group nl(S 3 -- Za) is isomorphic to the additive group Z[1/d] of rational numbers witk powers old in the denominator. Proof. -- Fundamental groups commute with inductive limits, so, by Proposition 4.4, nl(S s -- Za) is isomorphic to the inductive limit of zLzLz...o Remark. -- It is usually dangerous to speak of fundamental groups without spe- cifying a base point, but in this case the fundamental group is abelian, so there is no ambiguity. Knots and the mappings ~d,,. We Mll not need the following results in the sequel, but they may help the reader to understand why the mappings ~a,, are different. We will only discuss the case d = 2, but a similar discussion can be made for arbitrary d, and is a bit simpler in fact when d > 2. 24 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proposition 4.6. -- The solid tori "r~,~(T) are all unknotted if k = 0 and all knotted for n >i 2 if k # O, except that ~, _a(T) is unknotted. Proof. -- The case k = 0 is dealt with in Proposition 4.1. Next show that v~,~(T) is the (2, 2k -- 1) torus knot. This is genuinely knotted unless k = 0 or 1. Since ~"+~rT~,k ~ j is a companion of ~,k(T), this proves the result for all k except k = -- 1, which requires a separate argument. Observe that v~,k can be written v2,0 o w k, where w is the twist w(~, z) = (~, ~z). Then x~,~(T)= "r2,0ow~o'r2,0owk(T). The w ~ on the right can be ignored since T = w~(T) and since v2,0 extends to a homeomorphism of S a, the v~, o on the left can be ignored also. The result follows from the computation The mapping ~ ~-~ (~z, ~2k+1) is a parametrization of the (2, 2k + 1)-torus knot, which is indeed knotted unless 2k § 1 = 4- 1. To finish the proof, it must be shown that v~,_I(T ) is knotted. As above, "r~,_l(T ) = vz, o o w -1 o "r2, o o w -1 o "r2, 0 o w-I(T), and again ignore the w- ~ on the right and the %, 0 on the left. The reader may check that the core of the solid torus is then parametrized by ~ ~-~ (~4, ~-2 q_ r We leave it to the reader that this is a parametrization of the (2, -- 5)-torus knot. [] 5. The functions G mad q~ Recall the generalized H6non mappings of degree d, where a # 0 and the degree of q is less than d. Recall also the definitions of sets K and U i from the introduction. Looking at the formula for the H6non mappings, note that if x is reasonably large and large with respect to y, then the predominant behavior is that the x-coordinate gets raised to the d-th power. The following definitions are designed to state this rigo- rously. Set ~ to be at least as large as the absolute value of the largest root of Ix I - I q(x)l - (I a I + 2) x = 0. If p(x) = x2q - c, then the following value of ~ works: ~=~([a] q-2q-%/([a I +2) 2q-4[c[). HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I Define the regions V+, V_, and W C G ~ by V+ =((;)[ lyl~< Ix] and Ix[~> ~z}, w = lxl.< ~ and lyl.< ~ and Lemma 5.1. -- The sets V+ and V_ have the following properties: a) V+ C U and U+ = [.J,>~o F~ U_ = [-J.~>o F~ b) r(v+) cV+, F-I(V_) CV_; c) if (x,y) eV+, then [(F~ 2" for n= 1,2,... and if (x,y) eV_, then 1( F~ 1 + as n= 1,2,...; d) F(W) c W u V+; e) if (&y) eV_, then I F,(x,y)l < lYl and if (x,y) eV+, then I F~-a(x,y)l < Ix I" Proof. -- First consider the statements for V+. To see part b) let (x,y) ~ V+, and calculate: [p(x)-ay >>. ]p(x)]--]a[lyl>~ [p(x)]--[a[[x[>~ 2[x[. Thus F(V+) C V+ and for all n = 1, 2, ... Hence part r and also part a). Part d) is obvious since F(W)C{(x,y)l ly[< e}. Part e) is obvious also. For (x,y) ~ V_, the proofs are analogous using 1 1 [ a~l I P(Y) -- x [ >/~ I(P(Y) I - I x l) />~(IP(y)I-lyl)>~ 1+ lyl. [] Remark. ~ The proof of part c) of Lemma 5.1 shows that the first coordinates of an orbit starting in V+ grow at least geometrically. This is actually misleading. Since the dominant term of F1 is of degree d, the growth is like k a", Lemma 5.1 shows that every point eventually lands in V+ u W. 4 26 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proposition 5.2. -- There exist unique analytic functions q~+ : V 13- b suck that ep+ (;) -- oo in V+ or V_, respectively. and Proof. -- The function q~+ is constructed below and shown to have the required property. The proof for ~?_ is analogous. The proof of uniqueness will be given at the end of section 8. To simplify the formulas below set the notation x, = (F~ (x,y) and y, = (F~ (x,y). Note that x, is a polynomial in x and y of degree d" whose sole leading term is x a" and y, is a polynomial in x and y of degree d "- ~ whose sole leading term is x a"-~. To define q~+, meaning must be given to the limit ;) lim x lldn or rather the equivalent telescoping infinite product (;) xl/d lid n+l X . "1 Xn + l cO+ x ..... x~ an " " " Examine the individual terms of this infinite product: X~'l+11/dn+l -- [4 -~- q(x.) -- ayn] llan+l xlld n x~d n For (x,y) ~V+, F~ belongs to V+ and I q(x) - ay I q(x) l + l ly I ~< I q(x) l + I a II x I Ix I a -- 2 Ix ] ~< Ix Ix F <~ Ix I x 2 2 = 1 ixF_l~< 1 -- ~ -. Now, for the d"-th root use the principal branch of (1 + z) va". The infinite product converges as the series of the logarithms of the terms in the product converges. In the product above, consider the factor HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I The terms of highest degree in both polynomials involve only x. Since ]y [ ~< [ x [, the term (q(x,) -- ay,)/xa, is of order 1/x a". That is, there exists a constant C such that in V+ q(x,) -- ay, C which tends to 0 as (x,y) -+ oo in V+. Therefore the product is equivalent to its first term x. [] A refinement of this result will be needed, pushing the asymptotic development of ?:~ a bit further. We find it easiest to write = Uo(X) + ul(x)y + .... as a convergent power series in y, with coefficients Laurent series in x, which is clearly possible by the structure of V+. Proposition 5.4. ~ The following asymptotic, development holds: Uo(X) = X + O(Ixl) ana u (x) = dx _ 1+ o Proof. -- The development of u 0 is already in Proposition 5.2. From (5.3) above, the second and higher factors of the product cannot contribute larger terms than those given, and the first term gives the result. [] Proposition 5.5. -- The limits exist, are continuous on C ~', are pluri-harmonic on U + , and have the properties that Moreover, Proof. -- Again, the proof will be given for G+ and the proof for G_ is analogous. On V+, define G+ = log ] ~+ ]. Extend this definition to (x,y) ~ U+ as follows. By part a) of Lemma 5.1, there exists n > 0 such that F~ e V+. For such (x,y), define Further extend G+ to be zero on K+. 28 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH The definition is consistent: if a higher n had been used, then the result would be the same by Proposition 5.2 and clearly satisfies G+(F(x,y)) = dG+(x,y). The function G+ is harmonic on U+ since it is a real part of an analytic function on V+ and elsewhere the pullback of a pluri-harmonic function by an analytic mapping. It remains to see that it is continuous. Fix (x',y') ~J+. Then there exists N such that [[F~ ~ for all n/> N. For any M>N, there exists ~ so that if l[ (x",y") -- (x',y')II < =, then II F~ '') I[ < 0~. Note that F(W+) C W+ to V+. So the value of G+ on the first forward image of F~ '') which is in V+ is bounded by C = sup{G+(F(x,y))l(x,y ) eW}. So, G+(x",y")< C/d ~. [] Remark. -- The functions G~_ are obviously subharmonic. This fact has been observed by Bedford and Smillie ([BS2]) and by Fornaess and Sibony ([FS]). They use the fact that dd ~ G+ are positive (1, 1)-currents supported on J to derive analogs of the Brolin measure ([Br]) for Hdnon mappings. Fornaess and Sibony also prove that G are H61der continuous. 6. The global topology of H~non mappings The behavior of G+ is partially described by the following, in which solenoids make their first appearance in this subject. Theorem 6.1. -- The mapping G+ : U+ -+ R+ is a trivial fibration whose fibers are homeo- morph# to S 8 -- Y"a, o, embedded using the mapping "ca, o as in section 3. Proof. -- Represent the set U+(r) = G+l(log r) as the increasing union U+(r) = V+(r) to F-l(V+(r~)) to F-z(V+(r4)) to..., v+,., Jo+(;) / Proposition 6.2. -- a) For large s, V+ (s) is homeomorphic to a solid torus, and : V+(s) --+ { I I * I = s ) is a fibration with fibers homeomorphic to closed disks. b) The mapping G+ : V+ -+ R+ is a trivial bundle with fibers homeomorphic to solid tori above (R, oo)for R suffciently large. Proof of Proposition 6.2. -- For any z with ]zl~< 1 consider the function %(x) = 9+(x, zx). The function is defined and analytic for Ix[> 0r By Proposition 5.2, % has a simple pole at oo. The following lemma, which is an immediate consequence of Montel's Theorem, will be required. HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I Lemma 6.3. -- Let R > 1 ; then the space of analytic functions f: D -+ D~, satisfying 3,.(0) : 0 and f'(O) : 1 # compact. In particular, there exist numbers R1 and R e such that all such functions f are injective on D~I , and satisfy f(DR1 ) D D~. Applying Lemma 6.3 to ~+ (l/x, z[x)' which maps the disk of radius 1/, < 1 to the disk of radius 1, we see that there exist R x and R~ such that if ~ > R 2 and [ z ] ~< 1, then there exists a unique x such that ] x I > Ri and ~0+(x, zx) = ~. This shows that the mapping  : (x,y) ~ (~+(x,y),y/x) is a homeo- morphism V+(r) -+S i, � D for r~> Ri. [] (Proposition 6.2) To compute F in the coordinates (~, z), asymptotic developments of x and y as functions of ~ and z must be found. Proposition 6.4. -- The following asymptot# development holds: Proof. -- This is a standard inversion of an analytic function from Proposi- tion 5.3. [] (Proposition 6.4) Now compute F in the coordinates (~, z): (:) . \zx(~, z)] ~ x(~, z) p(x(~, z)) - azx(~, z)) Only the term ~a in the denominator contributes to the leading terms of the development of F, to give r = 1 +o + +o z+o(Izl) This mapping is not quite one of the va,~'s from section 3, but almost. Change variables once more, to (~, ~), where ~ = z~. In these coordinates, the following expres- sion holds: (6.5) F(~)= +o(1~1)+ d~_i+o ~-t-o(l~l) " ( t t tt ) In particular, it is conjugate to vd, o. 30 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Consider 0V+(r) with r sufficiently large so that Lemma 5.2 applies. Let y, be the curve parametrized by t ~-* (~ = e 2~", ~ = ~rd), 0 ~< t ~< 1, and let ~ be the curve parametrized by t ~ (~ = r, ~ = 0or a e2~t), 0 ~< t ~< 1. Proposition 6.6. -- If g : V+(r) -+ T is a homeomorphism with g(y,) a curve on T bounding a disk in S 3 -- T, then there exists a homeomorphism g' : V+(r a) -+ T such that the diagram V+(r) o > T V+(r d) -----+ T g' commutes. Moreover, g'(y,d) is again a curve which bounds in S 3- T. Proof of Proposition 6.6. -- The existence of g' and its uniqueness up to homotopy is an unpleasant topological generality. The substance of the proposition is in what g' does to y,d. The generality is a consequence of the following lemma. Lemma 6.7. -- Let X be a 2-sphere with three open disks with disjoint closures removed. Then the space of homeomorphisms of X mapping each boundary component to itself is contractible. Proof of Lemma 6.7. -- See [EE] and [Ha]. [] (Lemma 6.7) Both V+(r a) --F(V+(r)) and T- Va.o(T) are locally trivial fiber bundles over the circle with fibers homeomorphic to the sphere with three holes above. In each case, the functions called ~ are the fibrations. The following shows that these two spaces are fiber-homeomorphic. Cut the circle at some point, to manufacture two bundles V+ (r) and "r of spheres with three holes over the interval I. Both are trivial bundles, and hence homeomorphic toI xX. ~,~ Choose trivializations v:I x X~V+(r) and u:I x X4T. These induce monodromy mappings m.= (ul{1}xx)-' o Ul~o}xx and m.= (VI{1}XX)--I 0 VI{O} XX. The mapping u o v -1 would induce the desired homeomorphism V+(r) --->T if m~ o m; -1 were the identity. To arrange this, let m t be a family of homeomorphisms of X such that m0 =m~-Zom~ and m 1-=Id and define m:I � X-->I X X by re(t, x) = (t, mr(x)). If u is replaced by the trivialization uj = u o m, then the require- ment is satisfied. This manufactures a homeomorphism w = uz o v-Z:V+(r d) -- F(V+(r)) ~T -- vd, o(T). HI~.NON MAPPINGS IN THE COMPLEX DOMAIN. I 31 It is clear from the construction that its isotopy is unique (among fibered homeomor- phisms). It must be shown that w can be adjusted so as to coincide on 8(F(V+(r))) with za, 0 o g o F- 1 and that w maps ~,r to a curve on T which bounds a disk in S a -- T. Both questions are homotopy class questions: the second one obviously and the first because the restriction of w to the boundary can be adjusted to coincide with any homeomor- phism in its homotopy class. Both of these statements follow from the asymptotic expansion (6.5): + ~ + ~(~, n) where the error term is so small that if a parameter is put in front of it and varied from 1 to 0, then no homotopy classes are changed. Once the parameter is 0, the formula looks exactly like the formula for zd, 0" This is slightly misleading since ~ and B are in the circle of radius r and the disk of radius r, respectively, whereas the arguments of Cd, 0 are in the circle of radius 1 and disk of radius 2, respectively. We leave it to the reader to make the appropriate scaling after which the identity is a possible candidate for w. [] (Proposition 6.6) The proof of Theorem 6.1 is completed by induction. The same construction as above gives a sequence of homeomorphisms g', g", ..., where g(k) : V+(rdk) _+ T. Define G(k):F-k(V+(rak))---~'r~,ok(T) by G(k) = ~-ka, o o g(k~ o F k. That is, the following diagram commutes: 17+ (r) r ~ F-l(V+(rd)) r ~.F-~(V+(rd2)) c ~F-s(V+(rdS)) c :.... F2 1 F3 G " ~ d~ G " F~ d s " ~V+(r ) ) ,... 9 r_~,) a,u- * T. c g'[ ;- r~o3(T) c ..... ~T ~T ~... rd, O r~0 r~o In the end, U+(r)= [.J~~ is homeomorphic to IJ ~ s ~ k=o X~,ok(T) = _ Za, 0. This proves that the fibers of G+ are homeomorphic to S 8- Za, 0 for r sufficiently large. Proposition 6.6 admits parameters: if gt were a family of homeomorphisms as in the proposition, depending on a parameter t in an interval, then there exists g~ depen- I,V+(rg.,~[ 32 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH ding continuously on t and satisfying the conditions of the proposition. In particular, there exist homeomorphisms g, g', g", ... such that the following diagram commutes: U,~>~V+(r) ~> U,>RdV+(r ) ... T X [R, oo) ,d,% T X [R, oo) ... Applying this extension of Proposition 6.6 to the inductive proof above shows that the mapping G+ : U+ -+ R+ is a trivial bundle above (R, oo), with the same R as in Proposition 6.2. Now F ~ is a fiber homeomorphism of G+I((R/2 k, oo)) to G+I((R, oo)), covering x ~ 2 k x. Thus the mapping G+ is a trivial fibration over any compact subset of R+, hence locally trivial over R+, hence trivial since R+ is contractible. [] (Theo- rem 6.1) 7. The foHations of U+ The fibers of G+ are 3-dimensional manifolds, and not obviously objects of complex analysis. But because G+ is a pluri-harmonic submersion, U+(s) is naturally foliated by Riemann surfaces. We will show that every leaf is isomorphic to C and dense in U+ (s). The proof also shows that q~+ cannot be extended to all of U+. Lemma 7.1. -- Let W be open in C" and let h : W ~ R be a pluri-harmonic submersion. Set W(x) = h-l(x). Then each W(x) is a real (2n -- 1)-dimensional manifold, and it is naturally foliated by complex manifolds of dimension n -- 1, with tangent space at w E W(x) given by T~ W(x) tn iT,, W(x). Proof. -- Each W(x) is a manifold by the Implicit Function Theorem. The uni- queness of the foliation follows from the fact that a real hyperplane T of a complex vector space contains a unique complex hyperplane, namely T t% iT. The existence can be seen by setting locally h ---- Ref for some complex analytic function f, which is also a submersion, and observing that W(x) =f-l({ z ] Re z ---- x}) is naturally foliated by the fibers of f, which are complex manifolds of dimen- sion n- 1. [] Theorem 7.2. -- The leaves of the natural foliation 0fU+(s) are isomorphic to C and each is dense in U+ (s). Proof. -- Choose ~ e C -- D with [ E I - s, with s so large that Proposition 6.2 applies. The leaf through any point of ~+1(~) can be written ?+1(~) u F-~(?+*(~a)) u F-~(?+l(~a")) v... By Proposition 6.2, this is an increasing union of simply-connected surfaces, hence simply connected. HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I 33 V+ (s) . v+ (~) ~+ 'I C-,C ~ Co FIo. 7.1 F ~ (v+(s~)) /- F -s(v+(ss)) IOOOQ'~ FIo. 7.2 To see that this leaf is dense in U+(s), note that F-I(~I(~)) = U ~+I(o~) o~d= 1 (see Figure 7) and more generally F o-,(~u = ~U ~u 1(~). Since the d*-th roots of 1 are dense in the unit circle, each leaf is dense in V+ (s). Applying F repeatedly will make it dense in each term of the increasing union V+(s) kjF-I(V+(xd)) k./F-2(V+($a2)) kJ ..., which occurred in the proof of Theorem 6. I. It remains to show that the leaves are isomorphic to C. This requires the following proposition. 5 34 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proposition 7.3. -- Let X be a simply connected Riemann surface, and K C X a compact connected simply connected subset not reduced to a point. Suppose A0, A1, . .. is a sequence of disjoint annuli in X -- K such that the inclusion of A~ into X -- K is of degree 1. If oo mod(A~) = 0% if0 then the surface X is isomorphic to C. Proof. ~ The alternative is that X is isomorphic to D and X -- D is an annulus with finite modulus M. However, by the subadditivity of moduli of disjoint homotopic annuli ([A], [BH]), ~E~~ 0 mod(A~) ~< M. [] (Proposition 7.3) So find a sequence of annuli in a leaf with a divergent series of moduli. This is actually easy, as the annuli considered grow very rapidly. Consider the annulus For ~ sufficiently large, the function y/x is an isomorphism of A; onto the annulus 1/2 ~< [ z [ ~< 1 of modulus (log 2)/(2rc). The annuli A~, F-a(A;d), F-~(A;d,), ... have constant moduli. They are embedded in the leaf which contains q~+l(~), disjoint by Lemma 5.1, and embedded with degree 1 in the leaf with ?+1(~) removed. So the leaf is isomorphic to C by Proposition 7.3 (see Figure 7.2). This proves the result for [~1= s sufficiently large. The statement follows in general by observing that F maps bijectively leaves in U+(s) onto leaves in U+(sd). [] (Theorem 7.2) Proposition 7.4. -- The mapping induces a bijection of the set of leaves onto the (non-Hausdorff) group R/Z[1/d]. Proof. -- This was already shown in the proof of Theorem 6.2. [] Remark. -- There are analogous results for those in Sections 7, 8, and 9 if" F-1 ,, replaces " F ". HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I 35 8. An analytic description of U+ In this section the analytic structure of U+ is analyzed completely. This is done by showing that the " Riemann surface of ~+ " is a covering space of U+ isomorphic to (C -- D) � C. Therefore U+ is a quotient of (C -- D) � C by some discrete group of automorphisms, isomorphic to Z [2]/Z. The group of automorphisms of (C -- D) � C is infinite-dimensional, and since the covering group we are after is only defined up to conjugation, there is a good deal of freedom in the description. The particular choice is algebraically very pleasant, but may not be the best one from a dynamical point of view. The Riem~nn surface of ?+. Let U+ be the smallest quotient of the universal covering space of U+ on which ~+ is defined. This covering space should be thought of as the Riemann surface of ~+, but it cannot be defined as a subset of U+ � C since the fiber above a point of U+ is a coset of the group of dyadic angles, and hence not discrete in C, so the topology would be wrong. Being a covering space of an analytic manifold, U+ is a 2-dimensional complex manifold. The set V+ is naturally embedded as an open subset of U+, using the natural definition of V+ on V+, and of course there is an analytic function ~+ : ~+ -+ C -- which extends ? on V+. This mapping ~+ is a submersion, and its fibers are simply connected Riemann surfaces, hence isomorphic: to D, C, or the Riemann sphere. Theorem 8.1. -- The fibers of ~+ are isomorphic to C. Proof. -- This follows from Theorem 7.2. [] It is unfortunately not true that a 2-dimensional complex manifold with a sub- mersion to a subset of C and with fibers isomorphic to C is a locally trivial family of copies of C. Example. -- Let U ---- D � ~, where ~ is the Riemann sphere. Choose some non- analytic continuous mapping ,:D-+ C, such as ,(z) ----2. Consider the set U = U -- (graph of ~). The projection U -+ D does have all fibers isomorphic to C, but if it were analytically a locally trivial fiber bundle, then the section ~ would be analytic. However, with an extra condition, such submersions are locally trivial fiber bundles. Let X, Y be complex manifolds, and f: Y ~ X an analytic submersion. Let the vertical tangent bundle Ty/x =ker dr. Recall that a vertical 1-form is a section of ~om(Ty/x, C). Proposition 8.2. -- If all the fibers off are isomorphic to C, and if Y carries an analytic vertical 1-form ~, such that the integral of ~ along a path in one fiber vanishes only if the path is closed, then the mapping f: Y ~ X is a locally trivial fiber bundle. 36 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Proof. -- Choose any x ~ X, there exists a neighborhood U of x and a section ~: U ~ YoffNowdefineamappingg:f-l(U) -+ U � C bysendingy to (f(y), ~/~,, o~), where "t(Y) is a path in f-~(f(y)) joining r toy. There exists such a path since the fibers are connected, and the integral is independent of the choice since the fibers are simply connected, and all analytic 1-forms on a Riemann surface are closed. Clearly g makes the diagram f-'(u) uxc XZ , commute, and g is an isomorphism fiber by fiber. Indeed, the hypothesis implies that g is injecfive on each fiber, and an injective analytic mapping G ~ C is an isomorphism. [] Theorem 8.3. -- The projection ~+:U+ ~ G- D is a trivial analytic fiber bundle. Proof. -- By Cartan's Theorem B ([G]), it is enough to prove that it is locally trivial, since there are no topologically non-trivial affine-line bundles over C- D. Moreover, C -- D is a Stein domain, so the topological and the analytic classifications of such bundles coincide. Since log ?+ is well defined up to an additive constant, the 1-form co = d log ~+ is well defined on U+. Moreover, o~ has no zeros since any branch of log q~+ is a sub- mersion. Therefore one can locally find a function g on U+ such that dg A co --~ dx A dy. Let + be the restriction of ~'~_ dg to vertical tangent vectors. Since dg is well defined up a multiple of o~, this restriction gives a well defined vertical 1-form. To avoid conflict of notation with the exterior derivative set ~ = d in the following. Lemma 8.4. -- We have F* hb = (a/8) ~b. Proof. -- Clearly F*eo = 8o~, and F*dx ^ dy = a dx ^ dy. Thus up to multiples of co, F* dg= (a/~) dg. The result follows. [] (Lemma 8.4) Now to show that the criterion of Proposition 8.2 applies to +, project a curve in one fiber of U+ to U+. This projection will be closed only if the original curve was closed. Further take forward images of the curve until it lies in V+(r), for sufficiently large r. This will change the integral of + by dividing it by a power of a[8. So it is enough to show that for ~ sufficiently large, the integral over a curve y in q~+(~) vanishes only if 3" is closed. By Proposition 5.2, ~+ ,~ x, so that ,.~ dx/x, so that g can be chosen with dg ,-, x dy. Since the path y is nearly vertical, this term of dg contributes more than all other terms, and hence for such an integral to HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I 37 vanish, they-coordinates of the endpoints must agree. But this means that the endpoints agree, by Proposition 6.2. [] (Theorem 8.3) Next the structure of the group F C Aut U+ such that U+ = U+/r is examined. Proposition 8.5. -- The fibration ~J + ~ U+ induces an exact sequence of fundamental groups 0 , , 9 r , 0 >Z >Z Z >0 where I is represented by the canonical generator of hi(V+). Since V+ lifts to U+, this verifies that the left square is commutative, and the remainder follows. [] Proposition 8.6. -- There is a unique lift F ofF to ~J + mapping V + to V+, and it satisfies F(y(x)) ---- (8u F(x) for all y e F, where the composition law of F is written additively. Proof. -- Elementary covering space theory shows that the lift exists and is unique. The formula then comes from the fact that F : V+ --> V+ induces multiplication by on the fundamental groups. [] For the remainder of this section, let us restrict ourselves to degree 2, with p(x) = x ~ + c. It is possible to find similar formulas in higher degrees, but they require inverting a power series, and the computations are difficult and do not lead to simple expressions. Theorem 8.7. -- There exists a unique isomorphism ~J + ---> (C, -- I)) � G such that in that trivialization, the mapping F is written F(~, z) = ~2,~ z § ~. Proof. -- Choose a trivialization of the bundle U+ --> C -- D so that the zero section is tangent to a high order to the section s o : ~ ~ (x(~), O) at ~. There exists such a trivialization: in any trivialization, s o is a power series in ~ which converges in some neighborhood of oo. The sum s, of the first n terms of this series is an analytic section over all of C -- D and arbitrarily close to s o as n -+ oo. Now change trivializations so that s, becomes the zero section. Next an aymptotic expansion of F in this trivialization will be computed. 38 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH Lemma 8.8. -- If n is sufficiently large, then ff has an asymptotic expansion z) = z + + o(1) . Proof. -- Using a trivialization such that dz corresponds to +, then by Lemma 8.4, F* dz ---- (a/2) dz. This means that ff will act on each fiber as z v-+ (a/2) z + constant, and we are left with computing the constant (which depends on the fiber, i.e., on ~, of course). This " constant " can be understood as follows: in the chosen trivialization, take the point s,(~), and integrate qb from s,(~ 2) to F(s,(~)) along a path in the fiber. Return to the definition of de above. It was found from a function g on V+ satisfying dlog ~?+ ^ dg -~ dx ^ dy; so this integral is just g(F(s,(~))) -- g(s,(~)). In the formula above, replace s, by So, and only change arbitrarily small terms in the asymptotic expansion. Since q~+ is to first order x, setting g(x,y) ~ xy satisfies the equation d log q~+ ^ dg = dx ^ dy to first order. We invite the reader to check that ignoring the other terms of g will not affect the asymptotic expansion above. Setting (x(~), 0) = s0(E), compute g(x(~) 2 q- c, x(~) ) -- g(x(~2), O) = (x(~)" q- c) x(~) q- terms to be neglected. In Proposition 6.4, we started to compute x(~), but dit not quite go to the required precision. In fact, it would have been quite difficult to extract the relevant terms of the 0(1 ~ I) in arbitrary degree. The goal is the following formula: x(;) = +o , which we leave to the reader to verify. Then g(x(~)*-q-c,x(~))--g(x(~*),O)= ~--~-~ q- ~--~ q-0(1) 2~ q- 0(1). [] (Lemma 8.8) To complete the proof of Theorem 8.7, the o(1) above must be dealt with. This is some function v(~) on 13- D which vanishes at infinity. Making a change of trivia- lization (~, z) ~ (~, z + u(~)), the expression of F in the new tHvialization is c a HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I Next find u(~) so that v(~) -- ~ u(~) + u(~ 2) = 0. This can be done by formal power series, or by setting Clearly this series formally solves the problem and it converges since I v(~ 2~') [ ~< C I ~ ] -2m for some constant C. This shows the existence of the required trivialization. The uniqueness is clear from the uniqueness to the solution for u above. [] (Theorem 8.7) Finally the group P may be computed. Theorem 8.9. -- For each element j/2 k e r, there exists a unique polynomial P~,k(~) suck that in the trivialization above, the element of r corresponding to j]2 ~ is given by + p~,~(~)/" Proof. -- First, compute 7v2. By Proposition 8.6, c 9 z+~-~; This leads immediately to "~1/~(~, z) = -- ~, z + a More generally, suppose that ~'j/~k-t has been determined. Then Proposition 8.6 gives z+ ~ which can be rewritten +pj,~(~) + (e2"J/zk~) s ~ z + c 2 " z _ c e2,~j/~k = ~8 _ ~ +Pj-I,~(~ )~ This gives us pj,~(~). [] Finally, we fill in a gap in section 5, the uniqueness of q~ in Proposition 5.2. This actually requires knowing Theorem 8.7 for H6non mappings of any degree; the proof goes through with minor changes. 40 JOHN H. HUBBARD AND RALPH W. OBERSTE-VOKTH Proof of uniqueness for ~+. -- Suppose ~b : V ---> C -- D satisfies (+([;])t and First observe that + lifts to ~ : ~+ -+ C -- D. This is an application of the lifting cri- teflon for covering spaces. For all n, the space ~, = F-"(V+) has the homotopy type of a circle, and F : V, ~ ~,_ 1 induces multiplication by d on the fundamental groups. Thus + can be lifted recursively to all ~, by the formula and the proper choice of roots will guarantee that all lifts agree with t~ on V+. On each fiber of ~+, the function ~' must be constant, since it is a mapping C ~ C -- D. Thus we can write q~ = ~ o ~+, where 0~ : C -- D ~ C -- D is an analytic function; by restriction to V+, ~ = ~ o ~+. But a look at the functional equation shows that e must be of the form ~ ~-~ a~ k, with k a positive integer and a d- ~ = 1. Now the asymptotic expansion shows that a ----= 1 and k = 1. [] 9. The canonical compactificatlon of K+ Let C be the compactification of C adding a circle at infinity. Then any poly- nomial extends continuously to C and its restriction to the circle at infinity multiplies angles by the degree of the polynomial. This section contains a description of an analogous compactification C ~ of C 2, to which H~non mappings extend continuously. A 3-sphere is added at infinity and the mapping extends as the solenoidal mapping a on S 8. This further emphasizes the similar role which solenoidal mappings play for Hdnon mappings and angle doubling plays for quadratic polynomials. In particular, the closures of K_ and K+ in ~2 are the solenoids 2]+ and Z_ res- pectively (note the reversal). This sometimes allows us to measure " angles of external rays " in K_ in the solenoid Y,+. This will turn out to be important in the description of the topology of these sets. Theorem 9.1. -- There exists a compact Hausdorff space X homeomorphic to a closed four-baU with underlying set C a  S s such that a) the induced topologies on C a and S 3 are the standard topologies; b) C a is dense in X; c) the H~non mapping F extends to a homeomorpkism F : X ~ X; and d) the restriction of F to S 3 is the solenoidal mapping v o. HI'NON MAPPINGS IN THE COMPLEX DOMAIN, I Remark. -- This is a surprisingly difficult result to prove, considering that the analogous results have already been worked out at finite distance. The difficulty is that if a 3-sphere is added at infinity in the obvious way, with points corresponding to oriented directions in C 2, then all non-vertical directions are mapped to horizontal directions. In particular, the Hdnon mapping does not extend continuously, and even where it is defined, it fails to be injective. To make the extension of the Hrnon mapping injective, the horizontal (and vertical) directions will be examined with a microscope. More precisely, a delicate blowup of the circle at infinity in the x-axis will be made, replacing a point p by a way of approaching p, the method of approach which we focus on being the images of straight lines. Proof. -- Step 1. Blowing up a circle in S 3. Consider a compact differentiable curve F C S ~, and define the oriented blowup g~ of S 8 along r as follows. First choose a tubular neighborhood U of F such that there exists a unique geodesic of S ~ joining any point of U to F in U. For x ~ U, let ~ be the tangent vector at x to the geodesic joining x to F. Now define the blowup Ur as a subset of the unit tangent bundle TI(U ) to U C S 3 to be ~jr={(x,~) ~Tl(SS) l{~:k~f~176 is perpendicular to I ~ if x ~ r The obvious mapping ~ : Ur -+ U is an isomorphism on Ur -- ~-~(r), so glue S 8 -- F onto Ur to make ~. Above F, there is a torus, mapping to F as a bundle of circles. Step 2. A first microscope. Consider the solid torus T 1 parametrized by {(X,~) ~CZllX[ = 1,]~[~< 2}. The point (X, ~t) of this solid torus will be " at the end of " the ray t~ t~tj. Let F 1 be the curve of equation y = 0 in S 3. Glue 0T1 = { (~, ~) ] I X [ : 1, I ~ I = 2 } to 2"8 as follows: choose the circle in Ts above the point of F1 corresponding to ~- 1" 1 ~1"1 the point in the circle at infinity on the x-axis in the direction X. Radial projection of the curve t~..[ xt'] L r~tJ onto the sphere at infinity gives a curve which approaches the circle orthogonally in a definite direction. Identify (X, ~) with this direction. Let $3 be the 3-sphere blown up along r 1 with the torus T 1 attached as above. 6 42 JOHN H. HUBBARD AND RALPH W. OBERSTE-VOKTH Step 3. A second m~croscope. Unfortunately, distinguishing these " eventually horizontal curves " do not resolve the images of straight lines. Blow up the circle U S of equation Vt 2 = X in this torus further to see the constant term in a ray of the form More formally, consider the solid torus T2 I = 1, 14 la]). Glue T, to S~, by identifying the point (~, v) e 0T~ (i.e. I v I = [ a I) to the unit vector in the direction of v at (~, ~). Let S~ be S~ blown up along I' 2 with the torus T2 attached as above. Step 4. A topology on C 2 U ~s. A basis of neighborhoods of each point ofS s is needed. There is no difficulty at those points which correspond to points of S 3 not on the x-axis: take the cone over a neighborhood of such a point, and cut it off at some radius. It is not much harder to define a basis of neighborhoods of a point (X0, ~0) in Tt which is not on the circle F2 or on the boundary of T 1. Take a neighborhood v--((x, )llx-z01< 01< of (X, bt) e T1 and let the neighborhood consist of V and the points which can be written (Xt ~, Bt) for t > T and (X, ~) e V. An analogous description is possible for the points inside T 2. Given ~0 with ] t~o ] = 1 and v 0 with ] v o [ < [ a [, choose a neighborhood W of (b~0, v0) in the solid torus T2 defined by ]mu--tZo]<r o[<,, and a number T>0. Then a neighborhood of (~to, v0) will consist of W and the points of C 2 which can be written (~t 2 t 2, ~tt + v) with (~, v) E W and t> T. It is a good bit harder to define a neighborhood basis for a point on the boun- dary OT 1 or 0T 2. Let P be the solid paraboloid of points in C ~ which can be written (Xt 2,vtt) with [~t]< 2 (i.e. the set defined by the inequality ]y]2~< 4]xD. Choose (X0, ~t0) e 0Ta, i.e. ] X 0 I = 1 and [ ~0 ] = 2, and a neighborhood W1 of (X0, Vt0) in T 1. The intersection W 1 c~ 0T 1 corresponds to a set of unit vectors normal to r 1. Set W z to be the set of points in S 3 which are obtained by traveling a distance less than ~ from I~t on the geodesic tangent to such a normal vector. Now a neighborhood of (X0, ~t0) consists of Wt u W~, and the points which can be written (Xt ~, #) with (X, ~) e W 1 and t > T, and the points on rays through W2 of norm greater than T and outside the paraboloid P. We will leave to the reader the analogous construction for the boundary 0T2, as well as the proof of the following lemma. Lemma 9.2. -- The space C 9" U ~s with the topology above is compact Hausdorff. Step 5. The space C ~. The compacfification of C 2 constructed so far is adequate to do the HEnon mapping once in appropriate regions. It needs to be adapted to the HI~NON MAPPINGS IN THE COMPLEX DOMAIN. I 43 H6non mapping as a dynamical system. One way of doing this would consist of making an infinite sequence of " blow ups" as above, so as to resolve the images of parabolic rays, etc., and taking the projective limit. A different method will be used, inspired by the fact that a model for the locus at infinity already exists, as a dynamical system. Let S~ be a " new " copy of the 3-sphere, and ~ : S~ ~ S~ a solenoidal mapping. Let T' 0 C $1 s be a solid torus, such that T~ = a(T'o) is contained in the interior of To. The reason for this funny notation will become clear below. Back in 2 3, consider the solid torus T O of equation 15 [ ~< I x [. A ray in the cone over the boundary of T O can be written (t% t~) with [ ~ [ = [ ~ [ = 1. The image of such a ray is the parametrized curve t ~ (~a tz -t- c -- apt, ~t), which is asymptotic to the curve (9.3) s ~-~ ~a s z, ~s -t- ~ 9 This last curve converges as s ~ oo to a point in 0T2, and it is easy to see that the first is sufficiently close to the second so it converges to the same point. o o I -- P Choose a homeomorphism h:T 0 T 2 ~T o T2, conjugating the mapping : 0T 0 -+ 0T~ to the mapping T O ~ T 2 induced by the Htnon mapping as above. This is possible, by the classification of solenoidal mappings and the formula (9.3). Now put a topology on C ~ U S, 8 as follows. Attach T o --T' 2 to C 2 by k. For any point p of S~ -- (~+ u Z_), choose n such that ~(p) e T o -- T;, choose a neighborhood U of ~"(p) and define a neighborhood ofp to be F ~ c~ C 2) u ~~ n S~). This defines a neighborhood basis for all points in C 2 t_A SSl, except for those of the solenoids l~+ and Y~_. Recall that both solenoids are canonically homeomorphic to lira (S x, 2), and that there exists a unique mapping t~+ : T O ~ S 1 which semi-conjugates to angle-doubling (Theorem 3.1). A neighborhood of p = (..., Pz, Pl, P0) e Y~+ is the union of the set of (x,y) ~ C a such that F~ for all k~<N and ] v+(FO_k(x,y))[ --Pk < r for all k ~< N and the set of points p in n o -~ To k~N such that I t~+(a~ --Pk ] < r The union C a II $81, with this topology, is the space C 2. Step 6. Compactness of C a. It remains to verify that C a is compact, Hausdorff, and that the mapping F : C 2 ~ C 2 which is F on C 2 and ~ on S~ is continuous. 44 JOHN H. HUBBARD AND RALPH W. OBERSTE-VORTH To show compactness, take a sequence in C 2. Suppose that the sequence is in C ~ and that the norms of the elements (x.,y.) tend to infinity, as otherwise the sequence obviously has a convergent subsequence. Further, assume that the sequence lies in V+. Either there exists k and a subsequence (x.l,y.~) such that F~ r V+, or there isn't. In the first case, recall the set P from step 4, and choose k and a sub- sequence, which will still be called (x,l,y,i) such that FO-ktx ~,,)'nl/ . ~ V+ mp. If a subsequence can be chosen so that the rays through these points converge to a non- horizontal ray, then this subsequence converges to the point of S~ corresponding to this ray. If the rays through these points tend to the horizontal, then choose a subsequence so that the directions of the rays tend to I~1 on a curve orthogonal to I" 1. This direction is then a point of S~, which is the limit of the subsequence. This shows that the F ~ ~(x m,y./) have a subsequenee which converges, and hence so does (x,i,y,~), by the second part of sept 5. Now suppose that the number of times F-1 can be iterated on points of the sequence and stay in V+ tends to infinity. Then by the compactness of the circle and a diagonal argument, a subsequence of the (x,,y,) can be chosen so that the sequences ] 9+(V~ [ converge for all k, say to Pk. Clearly p~ = P~-I. Let p = (..., p~, Px, P0). The subse- quence clearly converges to p ~ Y~+. This shows compactness. Step 7. The space C 2 is I-Iausdor~. It remains to show that distinct points of C~ have disjoint neighborhoods. Clearly only points in S~ need to be considered, and Lemma 9.2 shows that only points in the solenoids need to be considered. Even here there is no problem. If p0 ~: P'o, then let s = ~ [Po -- Po ]- Then the s-neighborhoods of p and p' are disjoint. If p, = p~ for i < k and Pk ~e p~, then [ Pk -- P~ [ = 2 and so the ~-neighborhoods are disjoint. Since clearly the mapping if, given by F on C * and a on S s, is continuous (the bases of neighborhoods are invariant under F), this ends the proof of Theorem 9.1. [] Corollary 9.4. -- The closure of K+ in C 2 is K+ u ~_ , and the closure of K_ is K_ u ]g + . Proof. -- Clearly points of K and large norm are points where ~- 1 can be iterated many times staying in V+. Such points are in smaller and smaller neighborhoods of points of Y,+. [] 45 HI'NON MAPPINGS IN THE COMPLEX DOMAIN. I REFERENCES [A] AHLFORS, L., Conformal Invariants, McGraw-Hill, New York, 1973. [B] BEDFORD, E., Iteration of polynomial automorphisms of C 2, Proceedings of the International Congress of Mathe- maticians, 1990, Kyoto, Japan, Springer-Verlag, Tokyo, Japan, 1991, 847-858. [BSl] BEDFORD, E. and SMILUE,J., Polynomial diffeomorphisms of C~: currents, equilibrium measure, and hyper- bolicity, Invent. Math., 103 (1991), 69-99. [BS2] BEDFORD, E. and SmLLm, J., Fatou-Bieberbach domains arising from polynomial automorphisms, Ind/ana U. Math. J., 40 (1991), 789-792. [BS3] BEDFORD, E. and SmLLm, J., Polynomial diffeomorphisms of C ~, II: Stable manifolds and recurrence, J. Amer. 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