Real and Complex Dynamics of Unicritical Maps. Trevor Clark

Transcription

1 Real and Complex Dynamics of Unicritical Maps by Trevor Clark A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 2010 by Trevor Clark

2 Abstract Real and Complex Dynamics of Unicritical Maps Trevor Clark Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2010 In this thesis, we prove two results. The first concerns the dynamics of typical maps in families of higher degree unimodal maps, and the second concerns the Hausdorff dimension of the Julia sets of certain quadratic maps. In the first part, we construct a lamination of the space of unimodal maps whose critical points have fixed degree d 2 by the hybrid classes. As in [ALM], we show that the hybrid classes laminate neighbourhoods of all but countably many maps in the families under consideration. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps of degree d and allows us to transfer a priori bounds from the phase space to the parameter space. This result implies that the statistical description of typical unimodal maps obtained in [ALM], [AM3] and [AM4] also holds in families of higher degree unimodal maps, in particular, almost every map in such a family is either regular or stochastic. In the second part, we prove the Poincaré exponent for the Fibonacci map is less than two, which implies that the Hausdorff dimension of its Julia set is less than two. ii

3 Dedication To my Mom, Dad and Granny, who have always been there for me. iii

4 Acknowledgements First of all, I am deeply grateful to my advisor Mikhail Lyubich for many hours of inspiring conversations, his great patience and support over the years. I would also like to express my extreme gratitude to Artur Avila for several very helpful conversations. I would also like to thank Professors Roland Roeder and Michael Yampolsky and my fellow students Anna Miriam Benini and Davoud Cheraghi. I would like to thank Professor Jun Hu, for being my external examiner, and making several suggestions that improved the presentation of the results. For making my life much easier at times, I would like to thank Ida Bulat, Diana Leonardo and Marie Bachtis at the University of Toronto and Gerri Sciulli, Nancy Rohring, Lucille Meci and Barbara Wichard at Stony Brook University. I would like to thank Scott Sutherland and Alexander Kirillov for thinking of me when it came time to assign teaching jobs. I am grateful to my friends at the University of Toronto and Stony Brook University. Finally, I would like to thank Stony Brook University and the IMS for their hospitality, and NSERC for its support. iv

5 Contents 1 Introduction Holomorphic dynamics Regular or stochastic dynamics Past results Outline of the proof The critical exponent of the Fibonacci map History and Background Hausdorff dimension Conformal measures and equality of exponents Method Prelimaries General Notation Analysis Quasi-conformal maps Quasi-symmetric maps Holomorphic motions Quasi-conformal vector fields Equivariant vector fields Banach spaces v

6 2.3 Complex dynamics Polynomial-like maps Return maps A priori bounds Generalized polynomial-like maps Generalized polynomial-like families Unimodal maps Renormalization of unimodal maps Spaces of unimodal maps Real Puzzle Negative Schwarzian Derivative Quasi-polynomial maps A priori bounds Puzzle maps for maps with sufficiently big geometry Puzzle for maps with bounded geometry Hybrid Lamination Hybrid classes Infinitesimal theory A variational formula Macroscopic pullback argument Infinitesimal Pullback Argument Hybrid lamination for maps with sufficiently big geometry Splitting of the tangent space Hybrid lamination Hybrid lamination for maps with minimal post-critical set Splitting of the tangent space for generalized polynomial-like maps Tangent space for unimodal maps with minimal critical orbit vi

7 3.4.3 Infinitely renormalizable maps Lamination near maps with minimal post-critical set Regular or Stochastic Dynamics Parameter partition Sufficiently big geometry Maps with bounded geometry Slowly recurrent maps Conclusion Poincaré Exponents for Fibonacci Maps Preliminaries Quadratic Fibonacci maps Initial estimates Estimates for the Poincaré series for the return map Starting inside of the collar Starting outside of the collar Starting in precritical components Inductive Estimate for the Poincare Series Bibliography 97 vii

8 List of Figures 2.1 Puzzle map Fibonacci map viii

9 Chapter 1 Introduction This thesis has two parts. In the first part, we show that almost every map in an analytic family of unimodal maps with fixed degree d 2 is regular or stochastic, and in the second that the critical exponent of the Poincaré series for a quadratic Fibonacci map is bounded above by 2, which implies that the Hausdorff dimension of its Julia set is less than 2. The solution to each of these problems lies in holomorphic dynamics. 1.1 Holomorphic dynamics The study of iterations of rational functions began in the 1920 s with Fatou and Julia who initiated the study of the dynamics of rational maps f : Ĉ Ĉ. Even maps as simple as z z 2 + c can possess complicated dynamical behaviour caused by in part the simultaneous presence of both expanding features, for instance the growth of the degree of the map under 1

10 Chapter 1. Introduction 2 iteration, and contracting features, the presence of the critical point. Interest in the field was rekindled in the late 1970 s when computer pictures of Julia sets were observed. These pictures exhibit beautiful fractal structures that beg to be understood. Since then, the progress has been rapid, and numerous connections between holomorphic dynamics and other areas of math have been discovered. The Julia set of a rational map can be easily defined: it is the set of all z such that the family of iterates {f n } n=1 does not form a normal family near z. In the case of polynomial maps, this is equivalent to the condition that z be on the boundary of the set of points that escape to under iterations of f. The complement to the Julia set is called the Fatou set. Compared to the Julia set, the dynamics on the Fatou set are simple. Thanks to Sullivan s No Wandering Domain Theorem [Su3], they have been completely classified. In this thesis, we will be concerned with unicritical maps. Up to conjugation by an affine map, they are the functions f c : z z d + c. These maps possess at most one finite attracting or parabolic orbit. We call these maps hyperbolic or parabolic respectively. In addition to the Julia sets, another object that is of great interest in the study of unicritical maps is the Multibrot set, M d. These are the higher degree analogues of the Mandelbrot set, M 2. is the set of parameters c such that J(f c ) is connected. The degree d Multibrot set Looking at a picture of the Mandelbrot set, one immediately sees little copies of it everywhere inside of it. The idea of renormalization begins to explain this phenomenon. A map f c is called renormalizable if a high iterate of it possesses a restriction, Rf f p : U V, with U V, where U and V are open disks, that is a d-to-1 covering map (such a map is called polynomial-like) with a connected Julia set. To avoid tech-

11 Chapter 1. Introduction 3 nical difficulties, we will assume that all renormalizations are simple (this is a technical condition, that is automatically satisfied for real maps). This procedure gives us a map from the family {f c }, defined in a neighbourhood of c, to the space of polynomial-like maps and, by the Douady-Hubbard Straightening Theorem [DH], it specifies a little copy of the Mandelbrot set that contains c. If Rf is renormalizable, we can repeat this procedure. In case f is infinitely renormalizable, we can obtain a sequence R n f : U n V n of successive renormalizations, and consequently a nested sequence of little copies of the Mandelbrot set. When a map has no indifferent cycles, and possesses a recurrent critical point, but is non-renormalizable, it is possible to carry out a similar procedure known as generalized renormalization. It was introduced in [L3] and [LM]. The idea behind it is to study the dynamics of a map by studying the dynamics of the the return map to a small domain containing the critical point. We will call the resulting maps R-maps. We begin with a carefully chosen domain, V 0, and then consider the first return map to V 0, given by f 1 : V 1 i V 0. This is the first generalized renormalization of f. We let V 1 denote the component of the domain of the first return map that contains the critical point. The map f 2 : V 2 i V 1 is the second generalized renormalization of of f. Under certain conditions, it is possible to carry out this procedure indefinitely. We call the nested sequence of domains V 0 V 1 V 2... the principal nest. Rather than specifying a little copy of the Mandelbrot set, each generalized renormalization specifies a domain in the parameter plane, V k, containing c such that all maps in V k have the same combinatorics up to level k; that is, the same first landing maps to all levels of the principal nest up to V k ([L4], [L6]). The non-degeneracy of the annuli, V i \ V i+1 is of great importance. Through the quasi-additivity law and the covering lemma of Kahn and Lyubich, in all criticalities, it is known that both mod(v i \ V i+1 ) and (mod(v i \V i+1 ) possess combinatorially defined subsequences that are bounded away from 0 ([KL1], [KL2], [AKLS], [ALS]). Estimates of this nature are known as a priori

12 Chapter 1. Introduction 4 bounds. If f is a non-renormalizable quadratic map, then it is known that a certain subsequence of mod(v n \ V n+1 ), grows at least linearly ([L4]); however, in the higher degree case it may happen these moduli are bounded. Coping with this phenomenon is one of the themes of this work. 1.2 Regular or stochastic dynamics In [P], Palis conjectured that typical dissipative dynamical systems can be described by finitely many attractors, each supporting an ergodic physical measure, and that this description is robust under sufficiently random perturbations of the system. In a series of papers, [L6], [L8], [ALM], [AM1], [AM2], [AM3] and [AM4], the authors answer this question in full for generic families of unimodal maps. Central to their results is the fact that generic unimodal maps have non-degenerate critical points. This condition guarantees that certain geometric properties hold. We consider families of maps where the critical point has fixed degree d 2. Our goal is to prove the main theorems of [ALM], [AM3] and [AM4] in this setting. The starting point in the proofs of these results is the following theorem: Theorem (Theorem A). Every real hybrid class, Hf R, is an embedded codimensionone real analytic submanifold of U a. Furthermore, the hybrid classes laminate a neighbourhood of every non-parabolic map. The family U a is a space of unimodal maps and will be described precisely later. In the proof of this theorem, the main new difficulty we encounter is the presence of at most finitely renormalizable maps without decay of geometry, a phenomenon that does not occur for maps with a quadratic critical point; however, these maps possess generalized polynomial-like generalized renormalizations and can be treated in certain circumstances almost as if they were infinitely renormalizable. Once Theorem A is proved, we proceed by using the local lamination structure and renormalization arguments to parti-

13 Chapter 1. Introduction 5 tion the family according to combinatorics and transfer a priori bounds in the dynamical plane for at most finitely renormalizable maps to the parameter space. This puts us in a position to apply statistical arguments of [ALM], [AM2], [AM3] and [ALS] to prove: Theorem (Theorem B). Suppose that {f λ } is a non-trivial family of unimodal maps. Then almost every non-regular map in {f λ } is Collet-Eckmann and possesses a renormalization that is quasi-polynomial. By arguments in [AM4], Theorem B yields the following statistical description of typical non-regular parameters. Theorem (Theorem C). Suppose that {f λ } is a non-trivial family of unimodal maps with critical point of fixed degree d 2. Then almost every non-regular map, f λ, in the family satisfies: 1. The critical point is polynomially recurrent with exponent 1, i.e., lim sup ln f n λ (0) ln n = 1; 2. The critical orbit is equidistributed with respect to the absolutely continuous invariant measure µ, i.e., lim 1 n 1 φ(f i n λ(0)) = i=0 for any continuous function φ : I R; φdµ 3. The Lyapunov exponent of the critical value, lim 1 n ln Df n λ (f λ (0)), exists and coincides with the Lyapunov exponent of µ; 4. The multiplier of any periodic point p contained in supp µ is determined via an explicit formula by the itinerary of p and of the kneading sequence.

14 Chapter 1. Introduction Past results Let us describe the work done for maps with non-degenerate critical point. First, for the quadratic family, in [L6] it is shown that almost every non-hyperbolic real quadratic map that is at most finitely renormalizable possesses an absolutely continuous invariant measure using the analysis of the geometry and combinatorics of quadratic maps of [L4] and a criterion of Martens and Nowicki [MN]. Then, in [L8], Lyubich shows that the set of infinitely renormalizable maps in the real quadratic family has measure zero, and as a consequence establishes that almost every real quadratic map is regular or stochastic. Later, in [AM2], Avila and Moreira show that almost every non-hyperbolic real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit with exponent one. These theorems imply that typical quadratic maps have many other good statistical properties, for instance, exponential decay of correlations and stochastic stability. In greater generality, first, for families of quasi-quadratic maps, the regular or stochastic dichotomy was established in [ALM]. There, the authors show that this space of maps is foliated by connected codimension-one analytic submanifolds, the hybrid classes, and use them to transfer the regular or stochastic dichotomy in the quadratic family to any one-parameter, real analytic family that is not contained in a single hybrid class. In [AM3] the authors employ a generalization of the ideas of [ALM], which allows them to cope with the non-negative Schwarzian derivative case using a local argument, to show that their previous results for the quadratic family hold in any analytic family of unimodal maps with a non-degenerate critical point in which the hyperbolic parameters are dense. Following Avila and Moreira, we will call such a family non-trivial; for one-parameter analytic families that are not contained in a single hybrid class, this condition only fails when there is a persistent parabolic cycle, so the terminology is justified. The statistical argument of [AM2] depends on two phase-parameter relations whose definitions were motivated by the works of Lyubich in [L4] and [L6]: the topological phaseparameter relation and the phase-parameter relation. The topological phase-parameter

15 Chapter 1. Introduction 7 relation implies that nearby maps have topologically conjugate dynamics outside of a neighbourhood of the critical point and it gives a partition of parameter space according to the combinatorics of the landing map of the critical value to small neighbourhoods of the critical point. The phase-parameter relation gives a quantitative version of the topological phase-parameter relation for maps where the principal nest is eventually free of central returns. In [AM2], the existence of the phase-parameter relations depends on the global nature of quadratic maps, and the combinatorial theory of the Mandelbrot set. For general unimodal families it would be desirable to avoid using the particulars of the quadratic family as is done in the case of one-parameter families of quasi-quadratic maps by transferring estimates from the quadratic family to other families via the holonomy map along the hybrid classes. However, in these more general families, it is not known whether the hybrid classes are connected, and the lamination they yield is only local. In [AM3] the authors construct a special family, for which they know that the phaseparameter relations hold, that is tangent to a given family, and transverse to the local hybrid lamination. They then use the local lamination to show that the phase-parameter relations hold for the original family, and they use the phase-parameter relations to prove that almost every map in the family under consideration possesses a quasi-quadratic generalized renormalization. This puts them in a position to apply their statistical arguments from [AM2] to complete the proof that almost every map in the family is regular or Collet- Eckmann. The added technicality of the local argument not only makes it possible to avoid proving that the hybrid classes are connected, it also gives stronger results that could not be obtained by transferring estimates from the quadratic family, for instance, the polynomial recurrence with exponent one of the critical point. As is noted in Remark 2.1 of [AM2], the statistical analysis involved in the proofs of the theorems of that paper applies to any set of unimodal maps, {f λ } λ Λ, that are topologically conjugate to polynomial maps such that (1) for every λ Λ, f λ has a quadratic critical point and the first return map to a

16 Chapter 1. Introduction 8 sufficiently small nice interval has negative Schwarzian derivative, and (2) for almost every non-regular parameter λ, f λ has all periodic orbits repelling, is conjugate to a quadratic map whose principal nest is free from central returns, and the phase-parameter relations hold at all sufficiently deep levels of the principal nest. The work of Kozlovski in [K] allows us to take care of the requirement that return maps to small intervals have negative Schwarzian derivative. Moreover, when the principal annuli for a map f are sufficiently big, the existing arguments, with slight modifications, imply that the phase-parameter relations hold at f. Consequently, the key ingredients needed to remove the non-degeneracy condition on the critical point are: first, establishing that the set of infinitely renormalizable parameters has measure zero, second, that the set of at most finitely renormalizable parameters without exponential decay of geometry has measure zero and, finally, that around almost every non-regular point, there is decay in the moduli of parapuzzle annuli. In a forthcoming paper of Avila and Lyubich, [AL3], the theorem that the set of infinitely renormalizable parameters in the family z z d + c has measure zero is proved. This fact combined with arguments in [ALM] and [AM3] implies that in any non-trivial analytic family of unimodal maps the set of infinitely renormalizable parameters has measure zero. In this work, we will take care of the remaining two points Outline of the proof We begin by presenting background material in analysis and dynamics. The most important results presented are the complex a priori bounds of [KL2], [AKLS], [ALS], and the rigidity theorem of [KSS]. In Chapter 3, we develop the infinitesimal theory for the space of unimodal maps, and use it to construct the local laminations. In Chapter 4 we apply the lamination structure to construct a partition of the parameter space and conclude the proofs of Theorems B and C.

17 Chapter 1. Introduction 9 The majority of this part of the thesis is concerned with the proof of Theorem A. For maps with simple combinatorics, hyperbolic and Misiurewicz maps, the arguments are identical to those in the quadratic case, and so we will spend little time on them, likewise, for infinitely renormalizable maps. The remaining maps are those whose critical point is recurrent, but that are not infinitely renormalizable. We will consider the following two cases separately: the maps for which the geometry of the principal nest, I 0 I 1 I 2..., remains bounded, I n+1 / I n is never too small, and those for which the geometry is sufficiently big, I n+1 / I n is very small for some n. For a map f with sufficiently big geometry, the existing argument that the hybrid class of f is an embedded co-dimension one submanifold in U a requires only a few changes in the higher degree case. We begin by showing that the tangent space to a map f in the space of unimodal maps splits into the direct sum of the tangent space to the hybrid class of f and a vertical direction. Then, we show that the tangent space to the hybrid class is a co-dimension one subspace. Finally, with the infinitesimal preparation completed, we show that the hybrid class of f is in fact a co-dimension one submanifold. This is carried out exactly as in Section 8.2 of [ALM], and we will only describe it briefly. This final step is unchanged when we consider maps without sufficiently big geometry; however, the infinitesimal part of the argument is different. If f is a map with bounded geometry, the post-critical set of f is minimal. This allows us to use arguments like those of [LS1] and [LS3] to construct a persistent puzzle map given by a complexification of the first landing map under f to some sufficiently deep I n, and show that the associated return map possesses a generalized polynomial-like generalized renormalization. The use of this renormalization argument is particularly well suited to the study of the infinitesimal structure of the space of unimodal maps near such f. In [L7], Lyubich endows the space of polynomial-like maps with a complex analytic structure modelled on a family of Banach balls, and uses it to study the space of polynomial-like maps. Parts of this theory were first generalized to families of generalized

18 Chapter 1. Introduction 10 polynomial-like maps in [Sm]. Once we have proven some necessary results in the space of generalized polynomial-like maps, we use the renormalization operator to pull them back to the space of unimodal maps. While generalized polynomial-like maps are useful for us to treat the infinitesimal parts of our argument, in order to construct the lamination, we need to carry out certain approximation arguments in the space of unimodal maps. To do this, we require the existence of a persistent puzzle map as mentioned above. Even though the puzzles we construct for these maps do not have the good geometric properties of those that can be constructed for maps with sufficiently big geometry, the fact that they are persistent is sufficient for our purposes. Once we have constructed the lamination, we will be able to partition the parameter space according to combinatorics of the return map of the critical point back to deep levels of the principal nest and thus construct the principal nest of parapuzzle pieces. Moreover, we will be in a position to apply arguments of [AM3] to show that for parameters corresponding to maps with sufficiently big geometry this nest has a priori bounds. For maps without sufficiently big geometry, we use arguments that rely on the analyticity of the generalized renormalization operator to transfer a priori bounds from the parameter space for families of generalized polynomial-like maps back to the family under consideration. As soon as a priori bounds are obtained, we can apply a parameter exclusion argument of [ALS] to show that the set of parameters with exponential decay of geometry in the dynamical plane, decay of geometry in the principal nest of parapuzzle pieces, and whose principal nest is eventually free of central returns has full measure in the set of non-regular parameters. 1.3 The critical exponent of the Fibonacci map There are several notions of dimension of importance in holomorphic dynamics. Among them are Hausdorff dimension, hyperbolic dimension and the critical exponent of the

19 Chapter 1. Introduction 11 Poincaré series. The relationships among them are parts of a beautiful theory in measurable dynamical systems. Indeed, the interest in the Poincaré series, arises in part because of its application in the construction of conformal measures supported on the Julia set of a rational map and consequently its connections with the Hausdorff dimension of the Julia set. The Poincaré series with exponent δ of a rational map at a point x, not in the postcritical set, is defined by n=0 ζ f n (x) 1 Df n (ζ) δ. It can be shown that the Poincaré series with exponent δ converges or diverges independently of x, and we define the critical exponent to be the value δ cr (f) that separates the exponents for which the series converges from those for which it diverges. In this work, we will investigate the Poincaré series of the Fibonacci map and show that the critical exponent is less than 2. Theorem Suppose that f is the Fibonacci map. Then δ cr (f) < History and Background Hausdorff dimension Given a fractal subset of the plane, it is natural to study its Hausdorff dimension, and in particular, if its Hausdorff dimension is equal to or less than the topological dimension of the ambient space. Of couse, the Hausdorff dimension of a set in the plane cannot exceed two, and if it has positive Lebesgue measure, then the Hausdorff dimension equals two. However, the converse is not true, and it would be very interesting to characterize the maps whose Julia sets have Hausdorff dimension less than two. A great deal of work has been done on this problem. It is well known that the Hausdorff dimension of the Julia set of a hyperbolic rational map is less than two. Urbanski

20 Chapter 1. Introduction 12 proved that the same is true for those maps whose critical point in non-recurrent provided that their Julia set is not the whole sphere [U]. Initial progress on maps with a recurrent critical point was made by Przytycki. He showed that the Julia sets for maps satisfying the Collet-Eckmann condition and an additional technical assumption have Hausdorff dimension less than two in [P2]. In that paper, Przytycki introduced the method of shrinking neighbourhoods, which has been used to estimate the Hausdorff dimension of the Julia sets of rational maps satisfying various conditions of non-uniform hyperbolicity. In [Mc1], McMullen shows that the Julia set of a geometrically finite rational map (one with the property that every critical point in the Julia set has a finite forward orbit) has Hausdorff dimension less than 2. More generally, the authors of [GS]show that certain non-uniformly hyperbolic rational maps also have Julia sets with Hausdorff dimension less than two. The methods of this paper apply in our setting and imply that the Hausdorff dimension of the Julia set of the Fibonacci map is less than two. On the other hand, by a theorem of Shishikura ([S2]), it is known that the set of parameters c such that the Julia set of the map f : z z 2 + c has Hausdorff dimension two is generic on the boundary of the Mandelbrot set. Bishop and Jones ([BJ]) proved that the limit set of any geometrically infinite Kleinian group has Hausdorff dimension equal to two. Inspired by Sullivan s dictionary, it was conjectured that the same should be true for the Julia set of an infinitely renormalizable quadratic map. However, to the contrary, the authors of [AL3] prove that there exist infinitely renormalizable maps whose Julia sets have Hausdorff dimension less than two, and indeed that there exist infinitely renormalizable maps whose Julia sets have Hausdorff dimension arbitrarily close to 1 ([AL1]) Conformal measures and equality of exponents Motivated by the work of Bowen and Patterson, Sullivan applied the Poincaré series to construct δ cr -conformal measures supported on the limit set of a Kleinian group ([Su1])

21 Chapter 1. Introduction 13 and on the Julia set of a rational function whose Julia set is not the whole sphere ([Su2]). The connection between conformal measures and Hausdorff dimension can be seen in a theorem of Denker-Urbanski ([DU]): suppose that δ is the smallest number for which there exists a δ-conformal measure supported on the Julia set of a rational function, then δ = HD hyp (f). In case that f is a hyperbolic rational map, it is known that there exists a unique δ- conformal measure supported on J(f) and this measure is equivalent to the δ-hausdorff measure ([Su2]). Consequently for hyperbolic maps, all the quantities we have mentioned are equal; that is, HD(J(f)) = HD hyp (f) = δ cr (f) = δ (f). The equalities of these exponents are known in many other situations, for instance, for geometrically finite maps ([Mc1]) and more generally for non-uniformly hyperbolic maps ([P1], [GS]). For non-renormalizable quadratics, we have the following result of Przytycki: Lemma ([P1]). Suppose that g : z z 2 + c is a non-renomalizable quadratic polynomial with c outside the main cardiod of the Mandelbrot set, then HD(J(g)) = HD hyp (J(g)). Progress on this question has also been made in the case of infinitely renormalizable maps with a priori bounds by Avila and Lyubich ([AL3]). They show that for such maps HD hyp (f) = δ (f) = δ cr (f), and that when the Julia set of such a map has measure zero HD(J(f)) = HD hyp (f), and hence in this case all exponents are equal. Moreover, by applying a recursive quadratic estimate for the Poincaré series of an infinitely renormalizable map with a priori bounds and periodic combinatorics, they are able to establish that either HD(J(f)) = HD hyp (f) < 2, meas(j(f)) = 0 and HD hyp (f) = 2 or meas(j(f)) > 0 and HD hyp (f) < 2.

22 Chapter 1. Introduction Method To show that the critical exponent for the Fibonacci map is less than two, we inductively estimate the Poincaŕe series for the return maps to successively deeper levels of the principal nest. Using some estimates of [LM] and [AL3] we establish the basis for the induction. Roughly, we show that if we are at a sufficiently deep level n of the principal nest, then the δ-poincaré series of the return map to level n is small for some δ < 2. We pass to the level n + 1 of the principal nest through the first return map to level n followed by the landing map from level n to level n + 1. The proof works because the loss in the derivative caused by passing from level n to level n + 1 is recovered within two iterates of the return map to level n.

23 Chapter 2 Prelimaries 2.1 General Notation N = {1, 2,... } stands for the natural numbers; R stands for the real numbers; C stands for the complex plane and C stands for the Riemann sphere. D r (x) = {z C : z x < r}, D r = D r (0), and D = D 1. For an open set U C and a point z U, U(z) will denote the connected component of U containing z. For two sets X, Y C, we let dist(x, Y ) = inf x X, y Y x y. We let I denote the interval [ 1, 1], and [a, b] will denote the interval between a and b. For a > 0 let Ω a = {z C : dist(z, I) < a}. A topological disk is a simply connected domain in C and a Jordan disk is a topological disk bounded by a Jordan curve. A topological annulus is a doubly connected domain in 15

24 Chapter 2. Prelimaries 16 C. A set X C is called R-symmetric if it is invariant under the conjugacy map, z z. A function, vector field or differential defined on an R-symmetric set will be called R- symmetric if it commutes with the conjugacy. A set X is called 0-symmetric if it is invariant under z z. For a bounded function, vector field or differential, will denote its sup-norm. Given a bounded open set V C, B(V ) will denote the Banach space of bounded holomorphic functions f : V C which are continuous up to the boundary endowed with the sup-norm. The tangent space to a manifold M at a point x is denoted by T x M. For x X, orb(x) orb f (x) = {f n (x)} n=0 will denote the forward orbit or trajectory of x under f. In cases where the iterates are only partially defined orb f (x) consists of those points where f n (x) is well defined. ω(x) ω f (x) is the limit set of orb(x): ω(x) = n=0orb(f n (x)). A point x is called recurrent if x ω(x). Given a set Y X, the first return map to Y is defined as follows: for y Y, let F (y) = f l(y) (y), where l(y) N is the first iterate such that f l(y) (y) Y. Since such a moment may not exist, the first return map may only be partially defined; however, we will still write F : Y Y when it is convenient. The first landing map L : X Y between sets X and Y is the map L : x f l(x) (x) where l(x) N {0} is smallest number such that f l(x) (x) Y. Note that L Y = id and that this map may be only partially defined on X. Let X X and Y Y. Two maps f : X X and g : Y Y are called topologically conjugate or topologically equivalent if there is a homeomorphism h : X

25 Chapter 2. Prelimaries 17 Y such that h(x) = Y and h(f(z)) = g(h(z)), z X. (2.1.1) Classes of topologically conjugate maps are called topological classes. In case h defines a topological equivalence between maps f and g, we will say h is equivariant with respect to the actions of f and g. We will also use this terminology when h is only partially defined, in which case it means that (2.1.1) is satisfied whenever h is defined. Given a diffeomorphism φ : J J between two real intervals, its distortion or non-linearity is defined as sup x,y J log Dφ(x) Dφ(y). 2.2 Analysis Here we will collect the necessary facts from analysis that will be used later Quasi-conformal maps A homeomorphism h : U V between open sets U and V is called a quasi-conformal map, abbreviated qc map if it has locally square integrable distributional derivatives h, h, and h/ h k < 1 almost everywhere. Since this local definition is conformally invariant, we can define qc homeomorphisms between Riemann surfaces. One can associate with any qc map h, its Beltrami differential, which is defined by µ = h d z h dz, with µ < 1. We will identify the Beltrami differential of a qc map h : C C with h/ h.

26 Chapter 2. Prelimaries 18 With the analytic object, the Beltrami differential, we may associate a geometric object, a measurable family of infinitesimal ellipses, defined up to dilation, by pulling back the field of infinitesimal circles by Dh. The eccentricities of these ellipses are determined by µ (so they are uniformly bounded almost everywhere), and their orientations are determined by arg µ. The dilatation Dil(h) = 1 + µ 1 µ of h is the essential supremum of the eccentricities of these ellipses. A qc map h is called K-qc if Dil(h) K. A fundamental fact about qc maps is: Theorem (Weyl s Lemma). A 1-qc map is holomorphic. Quasi-conformal maps are much more flexible than conformal maps. In fact, any Beltrami differential µ with µ < 1 or, equivalently, any measurable field of ellipses with essentially bounded eccentricities, is given locally by a quasi-conformal map that is unique up to post-composition with a conformal map. Hence, any such Beltrami differential, defined on a Riemann surface S, a induces a conformal structure on S that is quasi-conformally equivalent to the original conformal structure on S. This and the Riemann Mapping Theorem imply: Theorem (Measurable Riemann Mapping Theorem). Let µ be a Beltrami differential on C with µ < 1. Then there is a qc map h : C C which solves the Beltrami equation: h/ h = µ. This solution is unique if it is normalized to fix three points in C. The normalized solution depends holomorphically on µ. The holomorphic dependence of h µ on µ in this theorem should be understood in the pointwise sense; that is, for any z C, the function µ h µ (z) mapping B 1 (L (C)) C is holomorphic.

27 Chapter 2. Prelimaries 19 A map f : U V between domains in C is called quasi-regular if it is the composition of a holomorphic map and a qc homeomorphism. Beltrami differentials can be naturally pulled back by quasi-regular maps µ f µ. Suppose that f : U V is quasi-regular. A Beltrami differential defined on an open set containing U V is called f-invariant if f µ = µ. Theorem (First Compactness Lemma). The space of K-qc maps h : C C fixing two points is compact in the uniform topology on the Riemann sphere. The image of an arc (a disk) under a qc map is called a quasi-arc (a quasi-disk). A subset of C is called qc-removable if any qc map H : C \ X C extends to a qc homeomorphism H : C C. Sets that are qc-removable have zero Lebesgue measure in the plane, and both quasi-arcs and points are qc removable Beltrami disks and paths Assume that a quasi-regular map f : U V admits an f-invariant Beltrami differential µ defined on the Riemann sphere. We can solve the Beltrami equation h λ h λ = λµ, λ < a 1 µ by means of qc maps h λ : C C fixing two given points. This gives us a family of maps f λ = h λ f h 1 λ each of which preserves the standard conformal structure and hence is holomorphic by Weyl s Lemma. The dependence of f λ on λ, given by the map (λ, z) f λ (z), λ < a, z h λ (U), is holomorphic. This family of maps is called the Beltrami disk through f in the direction µ. If we restrict λ to the real interval ( a, a), we obtain the Beltrami path through f in the direction of µ.

28 Chapter 2. Prelimaries Quasi-symmetric maps Let κ 1. A homeomorphism f : R R is called quasi-symmetric with constant κ if for any h > 0 and any x R we have 1 κ f(x + h) f(x) f(x) f(x h) κ. The dilatation, Dil(f), of a quasi-symmetric map is defined as the smallest such κ. The space of quasi-symmetric maps forms a group under composition, and the set of quasi-symmetric maps with constant κ that preserves an interval is compact in the topology of uniform convergence on R. Recall that quasi-symmetric maps are Hölder, but that they are not in general absolutely continuous. Quasi-symmetric maps are closely related to quasi-conformal maps by the following theorem. Theorem (Ahlfors-Beurling). Let f : C C be a K-qc homeomorphism preserving the real line. Then the restriction f R is κ(k)-qs. Vice versa, any κ-qs homeomorphism f : R R admits a K(κ)-qc extension to the complex plane. If X R and f : X R has a κ-qs extension to R, then we will say that f is κ-quasi-symmetric Holomorphic motions Given a domain Λ in a complex Banach space E with base point 0 and a set X 0 C, a holomorphic motion of X 0 over Λ is a family of injections h λ : X 0 C, λ Λ such that h 0 is the identity and h λ (z) is holomorphic in z. We let X λ X[λ] = h λ (X 0 ). The fundamental properties of holomorphic motions are given by the λ-lemma. It consists of two parts, one giving an extension of the motion and the other giving transversal quasi-conformality. The first extension result was obtained in [L1] and [MSS], and states

29 Chapter 2. Prelimaries 21 that any holomorphic motion of a subset of the Riemann sphere extends to a holomorphic motion of the closure. Indeed, by shrinking the base space, we have: Lemma (Extension Lemma [BR]). A holomorphic motion h λ : X 0 X λ of a set X 0 C over a Banach ball B r admits an extension to a holomorphic motion H λ : C C of the whole complex plane over the ball B r/3. For holomorphic motions defined over hyperbolic domains, we have an improvement on the Extension Lemma. Theorem (Slodkowski s Theorem [Sl]). If h λ : X 0 X λ is a holomorphic motion of a subset X 0 C over a hyperbolic domain D, then h λ admits a K(r)-qc extension to C, where r is the hyperbolic distance between 0 and λ. The second part of the λ-lemma is given by: Lemma (Quasi-conformality Lemma [MSS], [BR]). Let h λ : U 0 U λ be a holomorphic motion of a domain U 0 C over a hyperbolic domain D C. The maps h λ are K(r)-qc, where r is the hyperbolic distance between 0 and λ in D. Moreover, K(r) = 1 + O(r) as r 0. A holomorphic motion h λ : C C will be called normalized if it fixes the points 2 and 2. Since we will often be concerned with real restrictions of complex objects, it will be necessary for us to consider holomorphic motions that respect real symmetry. Assume that a complex Banach space E is endowed with an anti-linear isometric involution conj : E E. Let E R = {λ E conj(λ) = λ} and assume 0 E R. Let us say that a holomorphic motion of an R-symmetric set X C over B r is R-symmetric if h conj λ (z) = h λ (z). The Extension Lemma provides us with an R-symmetric extension of the holomorphic motion over B r/3.

30 Chapter 2. Prelimaries 22 Let µ λ, λ D, be a holomorphic family of Beltrami differentials on C such that µ λ < 1 for λ D and µ 0 = 0. Then the Measurable Riemann Mapping Theorem implies that there exists a unique normalized holomorphic motion h λ of C based at 0 such that µ hλ = µ λ. The converse is also true. Theorem (Bers-Royden). Let h λ be a holomorphic motion of an open set in C. Then µ hλ is a holomorphic family of Beltrami differentials. The compactness properties of qc maps also hold for holomorphic motions. Theorem ([D]). Let X C be a set containing 3 distinct points {a, b, c} and let Λ be an open subset of a separable Banach space. Consider a holomorphic motion of X over Λ as a map from Λ to the space of continuous maps from X to C endowed with the uniform metric. The space of all holomorphic motions h λ : X C, λ Λ, fixing {a, b, c} is compact in the uniform topology over compact subsets of Λ. Holomorphic motions of the plane over Λ can be viewed as complex codimension-one laminations on Λ C whose leaves are graphs of the functions λ h λ (z), z C. More generally, a codimension-one holomorphic lamination L on a complex Banach manifold M is a family of disjoint codimension-one Banach submanifolds of M, called the leaves of the lamination. Locally, the theory of codimension-one laminations is the same as the theory of holomorphic motions. The λ-lemma in the context of codimension-one laminations implies that the holonomy maps have quasi-conformal extensions and gives bounds on the dilatations of the extensions. Lemma Any codimension-one holomorphic lamination is transversally quasiconformal. We say that a holomorphic motion h λ : X 0 C is continuous up to the boundary if the map (λ, z) h λ (z) extends continuously to Λ X 0. An equipped tube h T is a holomorphic motion of a Jordan curve T. Its support is called a tube. A holomorphic

31 Chapter 2. Prelimaries 23 motion h λ of a Jordan curve T over Λ which is continuous up the boundary is called a tubing of T over Λ. The filling of a tubing T is the set U Λ C such that U λ is the bounded component of T \ T λ, λ Λ. A diagonal to a tubing is a holomorphic function ψ defined in a neighbourhood of Λ satisfying the following properties: (D1) For λ Λ, ψ(λ) belongs to the bounded component of C \ T λ, and for λ Λ, ψ(λ) T λ ; (D2) For any λ Λ, the point ψ(λ) has only one preimage γ(λ) T under h λ ; (D3) The holomorphic motion of a neighbourhood of γ(λ) in T admits an extension to a neighbourhood of λ; (D4) The graph of ψ crosses the orbit of γ(λ) transversally at ψ(λ); (D5) The map γ : Λ T has degree 1. Note that properties (D3) and (D4) imply that γ : Λ T is continuous so that (D5) makes sense. Given a set X 0 contained in the closed Jordan disk bounded by T and a tubing h over T, we say that a holomorphic, and continuous to the boundary, motion H λ of X over Λ fits to the tubing of T if for every λ Λ, we have H λ (z) = h λ (z) for z X 0 T, while H λ (z) / h λ (T ) for z X 0 \ T. Let h X T be a holomorphic motion such that motion of X fits to the tubing of T over Λ, and let φ be a diagonal of h T. By the Argument Principle, any leaf of h X intersects φ(λ) in a unique point with multiplicity 1. This allows us to define a map χ λ : X λ Λ such that χ λ (z) = w if (λ, z) and φ(w) belong to the same leaf of h. Each χ λ is a homeomorphism onto its image, if U X is open then χ λ U λ is locally quasi-conformal, and if Dil(h U) < then χ λ U λ is globally quasi-conformal with dilatation bounded by Dil(h U).

32 Chapter 2. Prelimaries 24 We call χ the holonomy family associated to the pair (h, φ). Let us summarize this discussion. Lemma ([ALS]). Let h λ : X 0 C be a holomorphic motion over a Jordan disk Λ continuous up to the boundary that fits to the tubing of a Jordan curve T. Let ψ be a diagonal to this tubing. Then for each point x X 0 there exists a unique parameter λ = χ(z) Λ such that h λ (z) = ψ(λ). The map χ : X 0 Λ is continuous and injective. Moreover, if z int(x 0 ) and h χ(z) is locally K-qc at z then χ is locally K-qc at z Quasi-conformal vector fields A continuous vector field v v(z)/dz on an open set U C is called K-quasi-conformal, abbreviated K-qc, if it has locally integrable distributional partial derivatives v and v, and v K. A vector field is called quasi conformal if it is K-quasi-conformal for some K. If µ L (C) one may obtain a qc vector field α satisfying α = µ. Local solutions to this problem are given by the Cauchy transform (see [AB]), 1 π µ(ζ) dζ d ζ, z ζ and global solutions are obtained from local ones. The Cauchy transform implies that the local solutions have modulus of continuity φ(x) = xln(x). Note that any two qc vector fields α and α satisfying α = α differ by a conformal vector field. A K-qc vector field will be called normalized if it vanishes at {2, 2, }. A continuous vector field v on a closed set X C is called quasi-conformal if it extends to a qc vector field on C. If a vector field admits a normalized qc extension to C, then we let v qc = inf β,

33 Chapter 2. Prelimaries 25 where β runs over all normalized qc extensions of v. Quasi-conformal vector fields are tangent at the identity to holomorphic motions, making them the infinitesimal counterpart to qc maps. Lemma ([ALM]). Let h λ : X C, λ D, be a holomorphic motion with base point 0. Then α d dλ h λ λ=0 is a qc vector field on X. Moreover if X is an open set, α = d dλ µ h λ λ=0 Many of the theorems about qc homeomorphisms have infinitesimal counterparts. In particular, for qc vector fields, we have the following compactness theorem. Theorem (Second Compactness Lemma). The space of K-qc vector fields on the Riemann sphere vanishing on three given points is compact in the topology of uniform convergence on C. Corollary ([ALM]). For any L > 0, there exists a C > 0 such that if α is a L-qc vector field on C that vanishes at and on the boundary of some interval T R, then α(z) < C T, for all z T Equivariant vector fields Let f : Ω C be a holomorphic map and let v be a holomorphic vector field on Ω. A vector field α is called equivariant on a set X Ω with respect to the pair (f, v) if for any z X, v(z) = α(f(z)) f (z)α(z).

34 Chapter 2. Prelimaries 26 This equation can be rewritten as f (α) α = v f. (2.2.1) This equation is obtained by linearizing the following commutative diagram: To see this we compute id+εα Ω Ω ε f C id+εα C. f+εv d (id + εα) f = d (f + εv) (id + εα) dε dε ε=0 ε=0 α f = f α + v Let X Ω and let α be a vector field on Y f(x). A vector field β is a called the lift of α to X by (f, v) if v = α f f β. This equation is obtained from the linearization of the following commutative diagram: X f id+εβ X ε f+εv C id+εα C. Later we will see that a necessary and sufficient condition for vector fields to be liftable is that v(0) must vanish to at least the same order as f (0). Also notice that a vector field is equivariant if and only if it is equal to its lift. Note that lifts preserve the qc-norm of vector fields: assume that the set X is open and that the vector field α is quasi-conformal. Let Y f(x). The lift β of α by (f, v)

35 Chapter 2. Prelimaries 27 can be written as β = f α v/f where v/f is holomorphic. So, β = (f (α Y )) = f (α Y ); notice that the first pullback acts on vector fields and the second acts on Beltrami differentials. Thus, β = f (α Y ) = (α Y ). Notice that the above calculation also implies that preserving the qc-norm of vector fields is the same as preserving the -norm of the corresponding Beltrami differentials Banach spaces Let E denote a complex Banach space. We say that a set K E\{0} is a cone if v K implies that λv K for all λ C\{0}. We say that a codimension-one subspace F is transverse to a cone K if F K =. Lemma ([ALM]). Let F be a codimension-one subspace transverse to an open cone K and v K. There exists C > 0 such that if c E \ K and w = w 1 + λv with w 1 F, then λ C w 1 E. For a sequence of subspaces F n E, let lim sup F n = Lin{v E lim inf dist E (v, F n ) = 0}. Lemma ([ALM]). If the F n are codimension-one subspaces, then lim sup F n is either E or a codimension-one subspace.