tspace.library.utoronto.ca · 2010-11-03 · Abstract Real and Complex Dynamics of Unicritical Maps...

Real and Complex Dynamics of Unicritical Maps

by

Trevor Clark

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

Copyright c© 2010 by Trevor Clark

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 dimen-

sion 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 Poincare exponent for the Fibonacci map is less than

two, which implies that the Hausdorff dimension of its Julia set is less than two.

ii

Dedication

To my Mom, Dad and Granny,

who have always been there for me.

iii

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

Contents

1 Introduction 1

1.1 Holomorphic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Regular or stochastic dynamics . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Past results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 The critical exponent of the Fibonacci map . . . . . . . . . . . . . . . . . 10

1.3.1 History and Background . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.3 Conformal measures and equality of exponents . . . . . . . . . . . 12

1.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Prelimaries 15

2.1 General Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Quasi-conformal maps . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Quasi-symmetric maps . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.3 Holomorphic motions . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.4 Quasi-conformal vector fields . . . . . . . . . . . . . . . . . . . . . 24

2.2.5 Equivariant vector fields . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.6 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

v

2.3 Complex dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Polynomial-like maps . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2 Return maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.3 A priori bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.4 Generalized polynomial-like maps . . . . . . . . . . . . . . . . . . 32

2.3.5 Generalized polynomial-like families . . . . . . . . . . . . . . . . . 35

2.4 Unimodal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.1 Renormalization of unimodal maps . . . . . . . . . . . . . . . . . 39

2.4.2 Spaces of unimodal maps . . . . . . . . . . . . . . . . . . . . . . . 40

2.4.3 Real Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4.4 Negative Schwarzian Derivative . . . . . . . . . . . . . . . . . . . 42

2.4.5 Quasi-polynomial maps . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4.6 A priori bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4.7 Puzzle maps for maps with sufficiently big geometry . . . . . . . . 45

2.4.8 Puzzle for maps with bounded geometry . . . . . . . . . . . . . . 47

3 Hybrid Lamination 53

3.1 Hybrid classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Infinitesimal theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 A variational formula . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.2 Macroscopic pullback argument . . . . . . . . . . . . . . . . . . . 56

3.2.3 Infinitesimal Pullback Argument . . . . . . . . . . . . . . . . . . . 57

3.3 Hybrid lamination for maps with sufficiently big geometry . . . . . . . . 62

3.3.1 Splitting of the tangent space . . . . . . . . . . . . . . . . . . . . 62

3.3.2 Hybrid lamination . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 Hybrid lamination for maps with minimal post-critical set . . . . . . . . 66

3.4.1 Splitting of the tangent space for generalized polynomial-like maps 66

3.4.2 Tangent space for unimodal maps with minimal critical orbit . . . 69

vi

3.4.3 Infinitely renormalizable maps . . . . . . . . . . . . . . . . . . . . 71

3.4.4 Lamination near maps with minimal post-critical set . . . . . . . 72

4 Regular or Stochastic Dynamics 73

4.1 Parameter partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.1 Sufficiently big geometry . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.2 Maps with bounded geometry . . . . . . . . . . . . . . . . . . . . 75

4.2 Slowly recurrent maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Poincare Exponents for Fibonacci Maps 83

5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Quadratic Fibonacci maps . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3 Initial estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4 Estimates for the Poincare series for the return map . . . . . . . . . . . . 92

5.4.1 Starting inside of the collar . . . . . . . . . . . . . . . . . . . . . 93

5.4.2 Starting outside of the collar . . . . . . . . . . . . . . . . . . . . . 94

5.4.3 Starting in precritical components . . . . . . . . . . . . . . . . . . 96

5.5 Inductive Estimate for the Poincare Series . . . . . . . . . . . . . . . . . 96

Bibliography 97

vii

List of Figures

2.1 Puzzle map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.1 Fibonacci map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

viii

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 Poincare 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 : C→ C.

Even maps as simple as

z 7→ z2 + 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

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 fn∞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

fc : z 7→ zd + 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, Md. These are

the higher degree analogues of the Mandelbrot set, M2. The degree d Multibrot set

is the set of parameters c such that J(fc) is connected. 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 fc is called renormalizable if a high iterate of it possesses a restriction,

Rf ≡ fp : U → V , with U b 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-


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 fc, 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 pro-

cedure. In case f is infinitely renormalizable, we can obtain a sequence Rnf : Un → 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 f1 : ∪V 1i → 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 f2 : ∪V 2i → 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, Vk, containing c such that all maps in Vk

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


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 1.2.1 (Theorem A). Every real hybrid class, HRf , is an embedded codimension-

one real analytic submanifold of Ua. Furthermore, the hybrid classes laminate a neigh-

bourhood of every non-parabolic map.

The family Ua 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 general-

ized polynomial-like generalized renormalizations and can be treated in certain circum-

stances 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-


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 1.2.2 (Theorem B). Suppose that fλ is a non-trivial family of unimodal

maps. Then almost every non-regular map in fλ 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 1.2.3 (Theorem C). Suppose that fλ is a non-trivial family of unimodal

maps with critical point of fixed degree d ≥ 2. Then almost every non-regular map, fλ,

in the family satisfies:

1. The critical point is polynomially recurrent with exponent 1, i.e.,

lim sup− ln |fnλ (0)|

lnn= 1;

2. The critical orbit is equidistributed with respect to the absolutely continuous invari-

ant measure µ, i.e.,

lim1

n

n−1∑i=0

φ(f iλ(0)) =

∫φdµ

for any continuous function φ : I → R;

3. The Lyapunov exponent of the critical value, lim1

nln |Dfnλ (fλ(0))|, exists and co-

incides with the Lyapunov exponent of µ;

4. The multiplier of any periodic point p contained in suppµ is determined via an

explicit formula by the itinerary of p and of the kneading sequence.


1.2.1 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 stochas-

tic 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 phase-

parameter relation and the phase-parameter relation. The topological phase-parameter


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 phase-

parameter 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 gen-

eralized 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, fλλ∈Λ, that are

topologically conjugate to polynomial maps such that

(1) for every λ ∈ Λ, fλ has a quadratic critical point and the first return map to a


sufficiently small nice interval has negative Schwarzian derivative, and

(2) for almost every non-regular parameter λ, fλ 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 7→ zd + 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.

1.2.2 Outline of the proof

We begin by presenting background material in analysis and dynamics. The most impor-

tant 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 .


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, I0 ⊃ I1 ⊃ I2 ⊃

. . . , remains bounded, |In+1|/|In| is never too small, and those for which the geometry

is sufficiently big, |In+1|/|In| 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 Ua 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

In, 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


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 param-

eter 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


Poincare series. The relationships among them are parts of a beautiful theory in measur-

able dynamical systems. Indeed, the interest in the Poincare 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 Poincare series with exponent δ of a rational map at a point x, not in the post-

critical set, is defined by∞∑n=0

∑ζ∈f−n(x)

1

|Dfn(ζ)|δ.

It can be shown that the Poincare series with exponent δ converges or diverges indepen-

dently of x, and we define the critical exponent to be the value δcr(f) that separates the

exponents for which the series converges from those for which it diverges.

In this work, we will investigate the Poincare series of the Fibonacci map and show

that the critical exponent is less than 2.

Theorem 1.3.1. Suppose that f is the Fibonacci map. Then δcr(f) < 2.

1.3.1 History and Background

1.3.2 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 Haus-

dorff dimension of the Julia set of a hyperbolic rational map is less than two. Urbanski


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 hyper-

bolicity. 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 7→ z2 + 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]).

1.3.3 Conformal measures and equality of exponents

Motivated by the work of Bowen and Patterson, Sullivan applied the Poincare series to

construct δcr-conformal measures supported on the limit set of a Kleinian group ([Su1])


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∗ = HDhyp(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)) = HDhyp(f) = δcr(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 1.3.2 ([P1]). Suppose that g : z 7→ z2 + c is a non-renomalizable quadratic

polynomial with c outside the main cardiod of the Mandelbrot set, then HD(J(g)) =

HDhyp(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

HDhyp(f) = δ∗(f) = δcr(f), and that when the Julia set of such a map has measure

zero HD(J(f)) = HDhyp(f), and hence in this case all exponents are equal. Moreover, by

applying a recursive quadratic estimate for the Poincare series of an infinitely renormaliz-

able map with a priori bounds and periodic combinatorics, they are able to establish that

either HD(J(f)) = HDhyp(f) < 2, meas(J(f)) = 0 and HDhyp(f) = 2 or meas(J(f)) > 0

and HDhyp(f) < 2.


1.4 Method

To show that the critical exponent for the Fibonacci map is less than two, we inductively

estimate the Poincare 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-Poincare 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.

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.

Dr(x) = z ∈ C : |z − x| < r, Dr = Dr(0), and D = D1.

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 ) = infx∈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

Ωa = 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

Chapter 2. Prelimaries 16

C.

A set X ⊂ C is called R-symmetric if it is invariant under the conjugacy map, z 7→ 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 7→ −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 TxM .

For x ∈ X, orb(x) ≡ orbf (x) = fn(x)∞n=0 will denote the forward orbit or trajectory

of x under f . In cases where the iterates are only partially defined orbf (x) consists of

those points where fn(x) is well defined.

ω(x) ≡ ωf (x) is the limit set of orb(x):

ω(x) = ∩∞n=0orb(fn(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 7→ 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 ′ →


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

supx,y∈J log|Dφ(x)||Dφ(y)|

.

2.2 Analysis

Here we will collect the necessary facts from analysis that will be used later.

2.2.1 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

∂h

dz

dz,

with ‖µ‖∞ < 1. We will identify the Beltrami differential of a qc map h : C → C with

∂h/∂h.


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 2.2.1 (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 2.2.2 (Measurable Riemann Mapping Theorem). Let µ be a Beltrami differ-

ential 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 hµ on µ in this theorem should be understood in the

pointwise sense; that is, for any z ∈ C, the function µ 7→ hµ(z) mapping B1(L∞(C))→ C

is holomorphic.


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 7→ 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 2.2.3 (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.

2.2.1.1 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

∂hλ∂hλ

= λµ, |λ| < a ≡ 1

‖µ‖∞

by means of qc maps hλ : C → C fixing two given points. This gives us a family of

maps fλ = hλ f h−1λ each of which preserves the standard conformal structure and

hence is holomorphic by Weyl’s Lemma. The dependence of fλ on λ, given by the map

(λ, z) 7→ fλ(z), |λ| < a, z ∈ hλ(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.


2.2.2 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 Holder,

but that they are not in general absolutely continuous.

Quasi-symmetric maps are closely related to quasi-conformal maps by the following

theorem.

Theorem 2.2.4 (Ahlfors-Beurling). Let f : C→ C be a K-qc homeomorphism preserv-

ing the real line. Then the restriction f |R is κ(K)-qs. Vice versa, any κ-qs homeomor-

phism 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.

2.2.3 Holomorphic motions

Given a domain Λ in a complex Banach space E with base point 0 and a set X0 ⊂ C, a

holomorphic motion of X0 over Λ is a family of injections hλ : X0 → C, λ ∈ Λ such that

h0 is the identity and hλ(z) is holomorphic in z. We let Xλ ≡ X[λ] = hλ(X0).

The fundamental properties of holomorphic motions are given by the λ-lemma. It con-

sists 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


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 2.2.5 (Extension Lemma [BR]). A holomorphic motion hλ : X0 → Xλ of a set

X0 ⊂ C over a Banach ball Br admits an extension to a holomorphic motion Hλ : C→ C

of the whole complex plane over the ball Br/3.

For holomorphic motions defined over hyperbolic domains, we have an improvement

on the Extension Lemma.

Theorem 2.2.6 (Slodkowski’s Theorem [Sl]). If hλ : X0 → Xλ is a holomorphic motion

of a subset X0 ⊂ C over a hyperbolic domain D, then hλ 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 2.2.7 (Quasi-conformality Lemma [MSS], [BR]). Let hλ : U0 → Uλ be a holo-

morphic motion of a domain U0 ⊂ C over a hyperbolic domain D ⊂ C. The maps

hλ 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 hλ : 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

ER = λ ∈ E|conj(λ) = λ

and assume 0 ∈ ER. Let us say that a holomorphic motion of an R-symmetric set X ⊂ C

over Br is R-symmetric if hconj λ(z) = hλ(z). The Extension Lemma provides us with an

R-symmetric extension of the holomorphic motion over Br/3.


Let µλ, λ ∈ D, be a holomorphic family of Beltrami differentials on C such that

‖µλ‖∞ < 1 for λ ∈ D and µ0 = 0. Then the Measurable Riemann Mapping Theorem

implies that there exists a unique normalized holomorphic motion hλ of C based at 0

such that µhλ = µλ. The converse is also true.

Theorem 2.2.8 (Bers-Royden). Let hλ be a holomorphic motion of an open set in C.

Then µhλ is a holomorphic family of Beltrami differentials.

The compactness properties of qc maps also hold for holomorphic motions.

Theorem 2.2.9 ([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 hλ : 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 7→ hλ(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 2.2.10. Any codimension-one holomorphic lamination is transversally quasi-

conformal.

We say that a holomorphic motion hλ : X0 → C is continuous up to the boundary

if the map (λ, z) 7→ hλ(z) extends continuously to Λ × X0. An equipped tube hT is a

holomorphic motion of a Jordan curve T . Its support is called a tube. A holomorphic


motion hλ 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 Uλ is the

bounded component of T \ Tλ, 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 \ Tλ, and for λ ∈

∂Λ, ψ(λ) ∈ Tλ;

(D2) For any λ ∈ ∂Λ, the point ψ(λ) has only one preimage γ(λ) ∈ T under hλ;

(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 X0 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 Hλ of X over Λ

fits to the tubing of T if for every λ ∈ Λ, we have Hλ(z) = hλ(z) for z ∈ X0 ∩ T , while

Hλ(z) /∈ hλ(T ) for z ∈ X0 \ T .

Let hX∪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 χλ : Xλ → Λ

such that χλ(z) = w if (λ, z) and φ(w) belong to the same leaf of h. Each χλ is a

homeomorphism onto its image, if U ⊂ X is open then χλ|Uλ is locally quasi-conformal,

and if Dil(h|U) <∞ then χλ|Uλ is globally quasi-conformal with dilatation bounded by

Dil(h|U).


We call χ the holonomy family associated to the pair (h, φ).

Let us summarize this discussion.

Lemma 2.2.11 ([ALS]). Let hλ : X0 → 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 ∈ X0 there exists a unique parameter

λ = χ(z) ∈ Λ such that hλ(z) = ψ(λ). The map χ : X0 → Λ is continuous and injective.

Moreover, if z ∈ int(X0) and hχ(z) is locally K-qc at z then χ is locally K-qc at z.

2.2.4 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)

z − ζdζ ∧ dζ,

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‖∞,


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 2.2.12 ([ALM]). Let hλ : X → C, λ ∈ D, be a holomorphic motion with base

point 0. Then

α ≡ d

dλhλ

∣∣∣∣∣λ=0

is a qc vector field on X. Moreover if X is an open set,

∂α =d

dλµhλ

∣∣∣∣∣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 2.2.13 (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 2.2.14 ([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 .

2.2.5 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).


This equation can be rewritten as

f ∗(α)− α =v

f ′. (2.2.1)

This equation is obtained by linearizing the following commutative diagram:

Ωid+εα−−−→ Ωεyf yf+εv

C id+εα−−−→ C.

To see this we compute

d

dε

∣∣∣∣∣ε=0

(id + εα) f =d

dε

∣∣∣∣∣ε=0

(f + εv) (id + εα)

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 = αf−f ′β. This equation is obtained from the linearization

of the following commutative diagram:

Xid+εβ−−−→ Xεyf yf+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)


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 dif-

ferentials. 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.

2.2.6 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 2.2.15 ([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 = w1 + λv with

w1 ∈ F , then |λ| ≤ C‖w1‖E.

For a sequence of subspaces Fn ⊂ E, let

lim supFn = Linv ∈ E| lim inf distE(v, Fn) = 0.

Lemma 2.2.16 ([ALM]). If the Fn are codimension-one subspaces, then lim sup Fn is

either E or a codimension-one subspace.


2.3 Complex dynamics

2.3.1 Polynomial-like maps

A holomorphic map f : U → V is called Douady-Hubbard polynomial-like if it is a degree

d branched covering between topological disks U, V with U b V . The fundamental

annulus of a polynomial-like map f : U → V is the annulus V \ U . In this paper, every

polynomial-like map will be assumed to be a unimodal map with its critical point at the

origin. We will also make the following technical assumptions:

• U is symmetric with respect to the origin and f is even,

• the domains U and V are bounded by piecewise smooth curves.

The symmetry assumption is just for convenience, and the assumption on the boundary

can always attained by shrinking the domain slightly.

Polynomial-like maps are considered up to affine conjugacy. We will say that a

polynomial-like map f is normalized at 0 if it is has the following expansion at 0:

f(z) = c+ zd +O(zd+1).

We let PL denote the space of unimodal polynomial-like maps and PL(µ) denote the

space of polynomial like maps f : U → U ′ with mod(U ′ \ U) ≥ µ.

A polynomial-like map is called real if the domains U and V are R-symmetric and f

preserves the real line; that is, f(U ∩ R) ⊂ R.

The filled Julia set of a polynomial-like map is defined as the set of non-escaping

points:

K(f) = z : fn(z) ∈ U, n = 0, 1, 2, 3, . . . .

Its boundary is called the Julia set, J(f) = ∂K(f). The setsK(f) and J(f) are connected

if and only if the critical point is non-escaping. Otherwise, J(f) = K(f) and this set is


a Cantor set.

If f is a real polynomial-like map with connected Julia set, then K(f)∩R is an interval

[−β, β], where β is a fixed point of f . Since f is considered up to affine conjugacy we

can also normalize it so that β = −1.

Two polynomial-like maps are called topologically equivalent if they are topologically

conjugate in some neighbourhood of their Julia sets. The hybrid class of a polynomial-like

map f is the set of all polynomial-like maps g for which there exists a qc homeomorphism

φ conjugating f to g with the property that ∂φ vanishes almost everywhere on K(f). If

f and g are in the same hybrid class, they are called hybrid equivalent. Note that when

f is hyperbolic (has an attracting cycle), the hybrid class of f consists of topologically

equivalent maps with the same multiplier of the attracting cycle.

By the Douady - Hubbard Straightening Theorem [DH], every hybrid class with

connected Julia set intersects the family

Pc : z 7→ zd + cc∈C

at a single point c of the Multibrot set Md, which is defined as the set of points c ∈ C

such that the Julia set J(Pc) is connected. With some extrar work to deal with parabolic

points, it follows that given a real polynomial-like map f , normalized so that β = −1 the

restriction f |I is topologically conjugate to a real unimodal polynomial map and that

this property holds for all maps in a neighbourhood of f .

Let us now consider the projectivized real tangent bundle L → C over the plane C.

An invariant line field on the Julia set J(f) is a measurable section X → L invariant

under the action of f , where X ⊂ J(f) is a measurable invariant set of positive planar

Lebesgue measure. Associated to an invariant line field on the Julia set, there is a family

of f invariant Beltrami differentials

µλ(z) = λe2iθ(z), |λ| < 1,


on X and extended by 0 to the whole complex plane. Hence, any invariant line field

generates a Beltrami disk fλ = hλ f h−1λ , |λ| < 1, of the map f where hλ is the

solution of the Beltrami equation ∂hλ = µλ∂hλ. This deformation is non-trivial on the

Julia set.

Lemma 2.3.1 ([LS2]). A real unimodal polynomial-like map with connected Julia set

carries no invariant line field on its Julia set.

So, since by [KSS] topological conjugacy of real maps implies qc conjugacy, the above

theorem implies:

Lemma 2.3.2. Let us consider two real non-hyperbolic unimodal polynomial-like maps

f and g with connected Julia set. If f and g are topologically equivalent, then they are

hybrid equivalent. Thus, there exists a unique unimodal polynomial in the topological

class of f .

2.3.2 Return maps

Let W be a quasi-disk and let Wj be a family of at least two quasi-disks inside W with

pairwise disjoint closures such that 0 ∈ W0. Assume additionally that

inf mod(W \Wj) > 0,

and, in case there are infinitely many Wj, that diam(Wj)→ 0.

An R-map (return map) is a holomorphic map f : ∪Wj → W such that for any j 6= 0,

f |Wj is a univalent map onto W , and f |W0 is a d − to − 1 covering of W branched at

0. Notice that by requiring there to be at least two disjoint quasi-disks inside W we do

not consider Douady-Hubbard polynomial-like maps as R-maps. When the number of

domains of an R-map f is finite, f is a unimodal generalized polynomial-like map.

We let W n = f−n(W ) and define the filled Julia set K(f) as ∩W n.


The components of f−n(W ) are called puzzle pieces of depth n. For x ∈ f−n(W ),

we let Y n(x) be the puzzle piece of level n containing x. Puzzle pieces containing 0 are

called critical, and we will denote the critical puzzle piece of level n by Y n

An R-map f is called renormalizable if there exists a puzzle piece V = Y n(0), n ≥ 1,

and an integer p > 0 such that V ⊂ fp(V ), the map fp : V → fp(V ) is a degree d map,

the puzzle pieces f j(V ) 1 ≤ j ≤ p are pairwise disjoint, and fmp(0) ∈ V, m > 0. The

map R(f) = fp|Y n(0) with minimal n as above is called the renormalization of f . It is

a unimodal polynomial-like map with connected Julia set.

We say that f has well defined combinatorics up to depth n if 0 belongs to the interior

of a puzzle piece of depth n. The combinatorics of f up to depth n, when it is well-defined,

is the label of the puzzle piece of depth n− 1 containing the critical value. In case f has

well defined combinatorics at all depths, we say that f is combinatorially recurrent if the

orbit of the critical point visits all critical puzzle pieces.

If f and f have the same combinatorics up to depth n, a pseudo-conjugacy (up to

depth n) between f and f is an orientation preserving homeomorphism H : (C, 0) →

(C, 0) such that H f = f H everywhere outside int Y n.

A critical puzzle piece Y n is a called a child of a critical puzzle piece Y m where

n > m if the map fn−m : Y n → Y m is unicritical with a degree d critical point. If f is

combinatorially recurrent, then every critical puzzle piece has a child. The children are

ordered by age; a child Y k is older than a child Y l if Y k ⊃ Y l, and thus k ≤ l. The first

child of a puzzle piece Y n coincides with the critical component of the domain of the first

return map to Y n.

A child Y q of Y v is called good if f q−v(0) is contained in the first child U ≡ Y u of Y v.

Suppose that f : U → V is a non-renormalizable R-map with recurrent critical point.

Then we can construct the principal nest for f , V ≡ V 0 ⊃ V 1 ⊃ V 2 ⊃ . . . where V i is

the first child of V i−1. We say that f has bounded geometry if there exists a constant C


such that

mod(V i \ V i+1) ≤ C.

Let f : ∪Wj → W and f : ∪Wj → W be two R-maps, and let h be a homeomorphism

of C equivariant of ∪∂Wj. If h(f(0)) = f(0), then for each j there is a homeomorphism

ψj : cl Wj → cl Wj coinciding with h on ∂Wj and such that h f = f ψj on Wj. Let

h1 =

ψj on Wj,

h on C \ ∪Wj.

Since diam Wj → 0, h1 is a homeomorphism of C. It is called the lift of h.

The definitions of topological equivalence, qc equivalence and hybrid equivalence are

the same as for polynomial-like maps.

2.3.3 A priori bounds

Suppose that f : ∪Uj → V is an R-map, and let V 0 ⊃ V 1 ⊃ V 2 ⊃ . . . be its principal

nest. Let Rn denote the return map to V n. If Rn(0) ∈ V n+1 we say that the return to

V n is central. Otherwise, it is called non-central. If a map is non-renormalizable, then

its principal nest has infinitely many non-central levels.

Theorem 2.3.3 ([ALS]). Suppose that V 0 ⊃ V 1 ⊃ V 2 ⊃ V 3 ⊃ . . . is the principal nest

for a non-renormalizable R-map f . Let nk → ∞ denote the levels of the principal nest

where there is a non-central return. Then there exists δ > 0 such that mod(V nk\V nk+1) >

δ.

2.3.4 Generalized polynomial-like maps

Definition 2.3.1. Let U = U0 ∪U1 ∪ · · · ∪Un be collection of disjoint simply connected

domains. A generalized polynomial-like map f : U → U ′ is a holomorphic map such that

f |Ui : Ui → U ′ is a di − to− 1 covering for each i.


Throughout, we will restrict to the case of a map

f :n⋃i=0

Ui → U ′

such that f |U0 is a d−to−1 covering of U ′ and for i 6= 0, f |Ui maps Ui univalently onto U ′.

Remark 2.3.1. Generalized polynomial-like maps possessing at least two connected com-

ponents in their domains are R-maps, so that the results stated for R-maps hold for these

maps as well.

2.3.4.1 Analytic structure on the space of generalized polynomial-like maps

Just as for polynomial-like maps, we can endow the space of generalized polynomial-like

maps with a complex analytic structure modelled on a family of Banach spaces. Suppose

that f : ∪jUj → V is a generalized polynomial-like map. Let Uf stand for the for family

of finite unions of topological disks with piecewise smooth boundaries on which f is a

generalized polynomial-like map. Let U = ∪Uj. If g ∈ BU(f, ε) is sufficiently close to

f in the Banach space B(U), then g is a generalized polynomial-like map defined on

a possibly smaller domain. Hence g represents a germ of a generalized polynomial-like

map, and we obtain an injection

jf,U : BU(f, ε)→ GPL.

This family of maps endows the space of generalized polynomial-like maps defined on

∪Uj with a complex analytic structure.

If the critical point does not escape, then the domains on which f is a generalized

polynomial-like map form a directed set, and we can identify the tangent space at f in


an appropriate space of generalized polynomial-like maps with the inductive limit

lim∪Uj⊂Uf

B∪Uj ,

which in turn may be identified with a family of finitely many vector fields vj such that

vj is defined on Uj, the vector field vanishes at −1, and v0 is normalized at the origin as

c+ azd+1 + . . . .

Let f : U0 ∪ U1 ∪ · · · ∪ Un → V be a generalized polynomial-like map and let U =

U0 ∪ · · · ∪ Un, where Ui b Ui are simply connected. Then, we say that the Banach

space Bnor(U) is compatible with f if whenever g : U0 ∪ · · · ∪ Un → V is a generalized

polynomial-like restriction of f with Ui ⊂ Ui, then K(g) = K(f). In this case, we say

that g is a representation of f subordinated to U . All representations of f have the same

filled Julia set and if the critical point does not escape, they have the same post-critical

set.

Definition 2.3.2. The tangent space to a generalized polynomial-like map f : U → U ′ is

the set of all vector fields on U normalized at the origin as v(z) = a0+a1zd+1+a2z

d+2+. . . .

Definition 2.3.3. The horizontal tangent space of f in Bnor(U) is the set of all holomor-

phic vector fields v ∈ TBnor(U) which admit a representation as

v = α f − f ′α

near the filled Julia set, where α is a qc vector field.

This set of vector fields is the tangent space to the hybrid class of f , and will be

denoted by Eh(f).

We say that a vector field v(z)/dz is vertical if there exists a holomorphic vector field

α(z)/dz on C \K(f) vanishing at ∞ such that

v(z) = α f(z)− f ′(z)α(z)


near the Julia set.

Suppose that f is a generalized polynomial-like map with non-escaping critical point.

Let B(f) be the space of f -invariant Beltrami differentials µ ∈ L∞ near K(f) such that

µ vanishes almost everywhere on the filled Julia set. Consider the Beltrami path, ht in

the direction of µ ∈ B(f); that is, the family of normalized solutions of the Beltrami

equations ∂ht/∂ht = tµ for small |t|. The velocity of this path at f ,

w =dhtdt

∣∣∣t=0,

is a vector field near K(f) that has locally square distributional derivatives and satisfies

the equation ∂w = µ. We let F(f) stand for the space of all such vector fields.

2.3.5 Generalized polynomial-like families

Let us consider a topological disk Λ ⊂ C with a basepoint λ0. Assume that fλ : ∪jUj[λ]→

V [λ] is a family of generalized polynomial-like maps such that the tubes ∪j∂Uj[λ] ∪

∂Vj[λ], λ ∈ Λ are all pairwise disjoint. Let h be a holomorphic motion of ∪Uj[λ0]∪V [λ0]

over Λ, such that fλ h = hf on the boundaries of the puzzle pieces. Moreover, assume

that for λ ∈ ∂Λ, fλ(0) ∈ ∂V [λ] and that as λ varies over ∂Λ the curve fλ(0) has winding

number 1 about 0.

Let V, ∪Uj denote the tubings of V, ∪Uj over Λ, and let

F : ∪Uj → V

denote the resulting tube map whose fibres are generalized polynomial-like maps.

It will be convenient to introduce some new notation. Let Y 0[λ] = V [λ], Y 1j [λ] =

Uj[λ].

Let us pull these tubes back by one iterate of f . For any λ ∈ Λ and any puzzle


piece Y 1j [λ] let W denote a component of the pullback of Y 1

j [λ] by f [λ]. Consider the

landing map Lj[λ] : W → Y 1j [λ]. We can now pull the holomorphic motion h back to the

boundary of W :

fλ h1(z) = h fλ0(z), z ∈ ∂W.

This holomorphic motion is an extension of h since it coincides with h on ∂Y1j . We will

keep the notation h for the extended motion.

Let W be a tube such that F(W) = Y1j , and extend the holomorphic motion to the

boundaries of these tubes by pulling it back by F .

Suppose that U ′[λ] is the puzzle piece containing f 2λ(0) and let Y 2

v [λ] denote the puzzle

piece (f |Y 1v [λ])−1(U ′[λ]).

Let us now define a new parameter domain Λ1 as the component of the set

λ : fλ(0) ∈ Y 2v [λ]

containing λ0. For λ ∈ Λ1 the maps fλ have the same combinatorics as fλ0 up to depth

2. Let us now define new tubes Y2j as the components of ∪k(f |Y1

k)−1(∪lY1

l ) over Λ1.

For λ ∈ Λ1 the critical value fλ(0) does not intersect the boundaries of the tubes Y2j ;

hence we can pull the holomorphic motion of ∂Y1j to a holomorphic motion over Λ1 of

the ∂Y2j that is equivariant on the boundary.

Suppose that Y 2j ⊂ Y 1

k . By construction, the motion of Y 2j fits to the tubing of ∂Y 1

k

over Λ1. As long as the orbit of the critical point is always in the interior of ∪Uj, this

process can be repeated infinitely many times giving us the Yoccoz puzzle pieces ∪Y nj

of any depth n and the nest of parameter puzzle pieces, parapuzzle pieces, Λn. Simply

by checking the properties in the definition, it is possible to prove that the critical value

map λ 7→ fλ(0), λ ∈ Λn is a diagonal to the tubing of ∂Y nv . Moreover, the puzzle pieces

of depth n+ 1 move holomorphically over Λn. Let us summarize this construction:

Lemma 2.3.4 ([ALS]). The set of parameters λ ∈ Λn+1 with the same combinatorics v


up to depth n is an open Jordan disk. The closure of the disk Λn is called the parapuzzle

piece of depth n with combinatorics Y n−1v . The boundaries of the puzzle pieces of depth

n provide us with tubings over Λn that fit to the tubings of the boundaries of the puzzle

pieces of depth < n containing it. The critical value map λ 7→ fλ(0), λ ∈ Λn is a diagonal

to the tubing of ∂Y n−1v .

Suppose that ij is the sequence corresponding to the central levels of the puzzle

defined so that V j = Y ij(0). Then we define V i = Λij . We will refer to the sequence

V1 ⊃ V2 ⊃ . . . as the principal nest of parapuzzle pieces.

Transferring the a priori bounds from the dynamical plane to the parameter plane we

have:

Lemma 2.3.5 ([ALS]). Suppose that fλλ∈Λ is a family of generalized polynomial-like

maps, and f ∈ fλ is a Yoccoz map with recurrent critical point. Let V1 ⊃ V2 ⊃ . . . be

the principal nest of parapuzzle pieces. Then there exists µ > 0 such that mod(V i\V i+1) ≥

µ.

2.4 Unimodal maps

Our reference for background in one-dimensional dynamics is the book of de Melo and

van Strien [MS].

A smooth map of the interval f : I → I is called unimodal if it has a single critical

point of even degree, and this point is an extremum. We will always assume that the

critical point is at the origin. Let cn = fn(0), n = 0, 1, 2, . . . ; and let Of denote the

closure of the post-critical set. Let U3 be the space of C3 unimodal maps f : I → I

with degree d critical point, that are even symmetric (so that f(−x) = f(x)) and endow

U3 with the C3 topology. We will normalize the maps so that −1 is a fixed point and

f(1) = −1. Moreover, we will assume that Df(−1) ≥ 1 for otherwise the map has −1 as

a global attractor or the map has a proper unimodal restriction to [f(1), 1] and a stable


fixed point in [−1, f(1)] (in case f(1) 6= −1). Note that in some cases we will consider,

it may happen that the map possesses attracting cycles that do not attract the critical

point. In this case, the map possesses a proper unimodal restriction that has at most

one attracting cycle which does attract the critical orbit (see [CE]).

Remark 2.4.1. The normalization and symmetry assumptions are made purely for con-

venience; indeed, through the argument of Appendix C in [ALM], all proofs generalize

to the case of non-symmetric maps.

Basic examples of unimodal maps are given by the unimodal polynomial maps

pτ : I → I, pτ (x) = τ − 1− τxd,

where d is even and τ ∈ [1/d, 2] is a real parameter.

Suppose that q is a periodic point of period n. We will let q = fk(q)n−1k=0 denote

its cycle. Let λ = (Dfn)(q) be its multiplier. The cycle q is called repelling if |λ| > 1,

parabolic if |λ| = 1, attracting if |λ| < 1, and super attracting if λ = 0.

The basin of attraction D(q) of an attracting cycle q is x ∈ I : fn(x)→ q. Similarly,

the basin of attraction for a parabolic cycle q is the set of all points that converge to q;

however, in this case, we omit preimages of q so that the basin of attraction will be an

open set.

A map f ∈ U2 is called Kupka-Smale if all of its periodic orbits are hyperbolic, and

f is called hyperbolic if it is Kupka-Smale and the critical point is attracted to a periodic

attractor. A hyperbolic map is called regular if its critical orbit is neither periodic nor

preperiodic.

The following is a corollary of a theorem of Mane, see [MS] p. 248.

Lemma 2.4.1 ([AM3]). Let f ∈ U2 and let T ⊂ I be a symmetric interval. If all periodic

orbits contained in I \ int T are hyperbolic, then

1. the set of points X ⊂ I which never enter int T splits into two forward invariant


sets: an open set U attracted by a finite number of periodic orbits and a closed set

K such that f |K is uniformly expanding and preperiodic points are dense in K;

2. there exists a neighbourhood V ⊂ U2 of f and a continuous family of homeomor-

phisms H[g] : I → I, g ∈ V such that g H[g]|I \ T = H[g] f , and H[f ] = id.

In our applications of this lemma, usually f will have all periodic orbits repelling,

and the forward invariant set will be the set of points which never land in a nice interval

containing the critical point; notice that this expanding set is usually a Cantor set.

2.4.1 Renormalization of unimodal maps

A unimodal map f is called renormalizable if there exists an interval J containing the crit-

ical point and an integer n ≥ 2 such that fn(J) ⊂ J and the intervals J, f(J), f 2(J), . . . , fn−1(J)

have pairwise disjoint interiors. The smallest such n is called the renormalization period.

Let n be the renormalization period and J 3 0 be the maximal periodic interval of

period n as above. This interval is bounded by a periodic point p and its symmetric

point. The restriction fn|J is called the prerenormalization of f

Let A : I → J be the affine rescaling mapping −1 to p. Then the map

R(f) ≡ A−1 fn A : I → I

is called the renormalization of f .

If R(f) is renormalizable, then the map f is called twice renormalizable, and contin-

uing in this way, we can define n times renormalizable maps for any n ∈ N ∪ ∞. It is

a well known fact that if a map is infinitely renormalizable, then its post-critical set is

minimal.

A hyperbolic or parabolic map with an attracting or, respectively, a parabolic cycle

of period n is at most finitely many times renormalizable for an infinitely renormalizable

map must have a recurrent critical point.


There are two types of renormalizations, and at times we will have to consider them

separately. A renormalization Rf ≡ fm|J : J → fm(J) of a renormalizable map f is a

satellite renormalization if there exist i, j, 0 ≤ i < j ≤ m− 1 such that f i(Dom(Rf)) ∩

(f j(Dom(Rf)) 6= ∅. On the other hand, if the sets (f i(Dom(Rf)), 0 ≤ i < m − 1 are

pairwise disjoint, then the renormalization is said to be primitive.

A unimodal map will be called Yoccoz if it is not infinitely renormalizable and has all

periodic orbits repelling. A Yoccoz map is called Misiurewicz if its critical point is not

recurrent.

2.4.2 Spaces of unimodal maps

Let a > 0. Recall that Ωa is the domain in the complex plane of points whose distance

from I is less than a. Let Ea ⊂ B(Ωa) be the complex Banach space of holomorphic

maps v : Ωa → C that are continuous up to the boundary, 0-symmetric and such that

v(−1) = v(1) = 0, endowed with the sup norm ‖v‖a = ‖v‖∞. It contains the real Banach

space ERa of real maps; that is, the subspace of Ea consisting of those maps that are

R-symmetric.

Let −1 ∈ Ωa be the constant function. We let Aa denote the complex affine subspace

−1 + Ea.

Let Ua = U3 ∩ Aa.

2.4.3 Real Puzzle

Suppose that f ∈ U2 is a finitely renormalizable Kupka-Smale map whose critical point

is recurrent, but not periodic. Then the first return map of f to its smallest restrictive

interval has an orientation reversing fixed point we will call α. A symmetric interval J

containing 0 is called nice in the sense of Martens if the orbits of its boundary points do

not intersect int(J).

The real Yoccoz puzzle PR for f is a collection of closed intervals P ni , n ∈ N ∪ 0,


called real Yoccoz puzzle pieces such that P 00 = [−α, α] and the P n

i , n > 0, are the

components of f−nP 00 . Intervals of the Yoccoz puzzle containing the critical point are

called critical and are labelled as P n0 . Every Yoccoz puzzle piece is nice. Moreover,

• any non-critical Yoccoz puzzle piece P ni is diffeomorphically mapped onto some

other puzzle piece P n−1k(i) ;

• any critical Yoccoz puzzle piece P n0 , n > 0, is folded into the Yoccoz puzzle piece

P n−11 containing the critical value c1 in such a way that f(∂P n

0 ) ⊂ ∂P n−11 .

Now, take a critical Yoccoz puzzle piece J0 ∈ PR and consider the first landing map

L to it; this map is called the real puzzle map associated to J0. The domain of this map

consists of a family J of disjoint Yoccoz puzzle pieces Ji ∈ PR, i ∈ N, satisfying

• any Ji, i > 0, is diffeomorphically mapped by f onto some other interval Jk(i) ∈ J ;

• there exists ni ∈ N such that the branch L|Ji = fni |Ji diffeomorphically maps Ji

onto J0.

The next well known lemma allows us to use nice intervals to study arbitrarily small

neighbourhoods around the critical point.

Lemma 2.4.2. Let f ∈ U2 be Kupka-Smale. If f is not hyperbolic and the critical orbit

is infinite, then for every ε > 0, there exists a nice interval [−p, p] ⊂ (−ε, ε) with p

preperiodic.

The definition of the principal nest for real maps is the same as the definition for

complex maps: to construct it we start with the nice symmetric interval, I0 = [−α, α],

and consider the sequence of puzzle pieces I0 ⊃ I1 ⊃ I2 ⊃ . . . , where I i+1 is the first

child of I i.

Let Rn be the first return map to In under iterates of f . Restricting Rn to the

components of the domain of the first return map that intersect the post-critical set we

obtain a generalized renormalization of f .


We say that a map f has sufficiently big geometry if there exists some n such that

|In+1/|In| < c, where c is a constant to be determined.

2.4.4 Negative Schwarzian Derivative

The Schwarzian derivative of a C3 map f : I → I is defined by

Sf =D3f

Df− 3

2

(D2f

Df

)2

in the complement of the set of critical points of f . Maps with negative Schwarzian

derivative enjoy many good dynamical and analytic properties. For instance, by a the-

orem of Singer, maps in U3 with negative Schwarzian derivative possess at most one

attracting periodic orbit, whose immediate basin of attraction contains either a critical

point or boundary point of I. Notice that our symmetry condition and derivative condi-

tion, Df(−1) ≥ 1 prevent the latter case from occurring. Real analogues of the Schwarz

Lemma and the Koebe Distortion Theorem also hold for maps with negative Schwarzian

derivative. As a consequence, by the work of Martens, away from the critical point, one

has excellent control of the distortion of maps with negative Schwarzian derivative. Be-

cause of the work of Kozlovski, [K], through (generalized) renormalization considerations,

we will be able to assume that the maps we are working with have negative Schwarzian

derivative. In [GSS], the authors use these ideas to prove:

Theorem 2.4.3. If f is a C3 unimodal map with a non-flat and non-periodic critical

point, there exists a nice interval J containing the critical point and a real-analytic dif-

feomorphism g : J → g(J) ⊂ R that conjugates the first return map to J to a function

with negative Schwarzian derivative.

The following theorem allows us to remove the negative Schwarzian derivative con-

dition for at most finitely renormalizable maps. It follows immediately from Theorem

2.4.3 and Lemma 2.4.1.


Lemma 2.4.4 ([AM3]). Suppose that f ∈ U3 is at most finitely renormalizable. There

exist i > 0, an analytic diffeomorphism s : I → I and a neighbourhood V ⊂ U3 of f , such

that there exists a continuation I i[g], g ∈ V of I i (H[g](I i) = I i[g]) such that the first

return map to s(I i[g]) by s g s−1 : I → I has negative Schwarzian derivative.

In the case of infinitely renormalizable maps we have the following well known esti-

mate:

Lemma 2.4.5. If f ∈ U3 is infinitely renormalizable, then if T ⊂ I is a small enough

periodic nice interval, the first return map to T has negative Schwarzian derivative.

2.4.5 Quasi-polynomial maps

A map f ∈ U3 is called quasi-polynomial if any nearby map g ∈ U3 is topologically

conjugate to a unimodal polynomial map. We let U ⊂ U3 denote the set of quasi-

polynomial maps. By the work of Milnor and Thurston, and the No Wandering Intervals

Theorem, a map f ∈ U3 with negative Schwarzian derivative and D2f(−1) < 0 is quasi-

polynomial so unimodal polynomials are quasi-polynomials.

Together with with the work of Kozlovski, similar ideas lead to the following criterion

for a map to be quasi-polynomial.

Theorem 2.4.6 ([AM3]). Let f ∈ U3. If f is not conjugate to a unimodal polynomial,

then there exists a (not necessarily hyperbolic) periodic orbit, which attracts an open set.

In particular, if all periodic orbits of f are repelling, then f is a quasi-polynomial map.

A quasi-polynomial map is called hyperbolic or regular if it possesses an attracting

periodic orbit. In this case, the orbit of the critical point converges to the attracting

cycle, and so such a map cannot possess more than one attracting cycle. Moreover, if it

has one, then almost all orbits converge to it.

A quasi-polynomial map f ∈ U is called parabolic if it possesses a parabolic cycle.

Just as for hyperbolic maps, in this case, the critical point is contained in the basin, D(q),


of the parabolic orbit, and the basin has full Lebesgue measure in I. Hence, we have

that such a map can possess at most one parabolic cycle and cannot be simultaneously

parabolic and hyperbolic.

2.4.6 A priori bounds

In the study of maps with a recurrent critical point, many nice properties follow from

there being definite space between nested puzzle pieces. For non-minimal maps, the proof

of the following theorem is unchanged in the higher degree case.

Theorem 2.4.7 ( [L3]). Suppose that f is a Yoccoz map whose critical orbit is non-

minimal. Then f has sufficiently big geometry.

For infinitely renormalizable maps, the connection between real analytic unimodal

maps and polynomial-like maps comes from the fact that sufficiently high renormaliza-

tions of unimodal real analytic maps are unimodal polynomial-like maps.

Theorem 2.4.8 ([LS1]). Let f be a real analytic unimodal infinitely renormalizable map

and f s : V → V be an arbitrary renormalization of this map to some periodic interval

V ⊂ R containing c. Then this map has a polynomial-like extension F : Ω0 → Ω with

• the modulus of Ω \ Ω0 is bounded from below by δ;

• the diameter of Ω is at most C times the diameter of V .

Here d depends only on the degree of the map, and C is universal. In fact, δ is asymptoti-

cally like const/d as d→∞. Moreover, if f is not s/2 renormalizable, then Ω0∩ω(c) ⊂ V .

Note that this property is robust; that is, if it holds for some map f0 ∈ U then it

holds with the same n for nearby maps.

We will treat the final case of at most finitely renormalizable maps with minimal

post-critical set in Section 2.4.8.


2.4.7 Puzzle maps for maps with sufficiently big geometry

The main tool we use to analyze the dynamics of Yoccoz maps are puzzle maps. The

domain of a puzzle map is the complexification of a real puzzle for a map f . A component

of the domain of a puzzle map can be modelled by the union of symmetric disk sectors:

given 0 < θ ≤ π/2, and A ⊂ R, let Dθ(A) be the intersection of two round disks D1

and D2 where D1 ∩ R = A, ∂D1 intersects R at the angle θ and D2 is the reflection of

D1 about the real axis. While the non-linearity of f prevents the description from being

as simple as this, when the geometry is big, the distortion of f is small making this a

reasonable picture to keep in mind.

Let us fix a deep level n of the principal nest for f such that |In|/|In−1| is small and

let A0 = In. Then A0 is a nice interval for f and we may associate with it the real puzzle

Aj.

Lemma 2.4.9 (Lemma 5.5 [ALM]). Let 0 < φ < ψ < γ < π/2 be fixed. For arbitrarily

large k > 0, if |In|/|In−1| is small enough, there exists a sequence of open Jordan disks

such that Dφ(Aj) ⊂ V j ⊂ Dψ(Aj) and V 0 = D(φ+ψ)/2(A0) with the following properties:

(1) if j 6= 0 and f(Aj) ⊂ Ak, then f : V j → V k is a diffeomorphism;

(2) if f(A0) ∩ Aj 6= ∅, then mod(f(V 0) \Dγ(Aj)) > k.

A map f with an associated puzzle as in this lemma will be called a geometric puzzle

map.

Suppose that f is a regular map, and let A denote the set of all attracting periodic

orbits of f and let B = x ∈ I : fn(x)→ A denote the basins of the attracting orbits.

Definition 2.4.1. Suppose that f is a Kupka-Smale map. We say that a homeomorphism

h : I → C is f -admissible if the following holds. Let T be a periodic component of B \A

which does not contain 0, and, writing T = (a, b) with |a| < |b|, we have that the interval

[−a, a] is nice. Then h takes d = (a + b)/2 to h(d) = (h(a) + h(b))/2 and h|[d, f q(d)] is

affine, where q is the period of T .


Let V ⊂ Aa be a real-symmetric neighbourhood of f . We will say that the puzzle

persists in V if there exists a real-symmetric holomorphic motion h over V given by a

family of transition maps hg : C→ C, g ∈ V such that:

(1) hg|C \ Ωa = id;

(2) g hg|V \ V 0 = hg f, g hg|∂V 0 = f ;

(3) hg|I is f -admissible and g hg|([−1, r] \ V ) = hg f .

The last condition in this definition defines hg uniquely in [−1, r] \ V .

Lemma 2.4.10 (Lemma 5.6 [ALM] and see Appendix A of [AM3]). Let f ∈ Ua be a

Yoccoz map. If |In|/|In−1| is sufficiently small, then there exists neighbourhood of f where

the puzzle persists.

2.4.7.1 Holomorphic motions of puzzle maps

Let A1 = [l, r], with l < r, denote the real puzzle piece containing the critical value, and

N = [−l, l]. The components of Aj that do not intersect A1 ∪ N will play no role in

what is to follow, and we will discard them without changing notation.

Let Z = Dγ(A1) ∪ Dγ(N), and let Υ be the space of holomorphic vector fields on

Z whose first derivatives extend continuously to Z and which vanish up to the first

derivative on fn(∂A1), n ≥ 0, with the property that v|Dγ(N) is an odd symmetric

vector field. Let Υ = Υ1 ⊕ Υ2, where a vector field v ∈ Υ1 if v|Dγ(N) = 0, and v ∈ Υ2

if v|Dγ(A1) = 0. Let fv = f (id + v).

Lemma 2.4.11 ([ALM],[AM3]). There exists ε > 0 such that if v belongs to v ∈ Υ :

‖v‖ < ε, then there exists a holomorphic motion hv over D, which is real-symmetric if

v is real-symmetric, and is defined by a family of transition maps hv[0, λ] ≡ hvλ : C →

C, λ ∈ D such that:

(1) hvλ|C \ Z = id, hλv |∂f(V 0) = id;


Ui

U0

f

0

fn0 (U0)

Figure 2.1: Puzzle map

(2) fλv hvλ|V \ V 0 = hλ f, fλv hvλ|∂V 0 = f .

The holomorphic motion hv of the above lemma will be said to be compatible with v.

2.4.8 Puzzle for maps with bounded geometry

Let U ⊂ C be a bounded open set. We say that a holomorphic function f : U → C

belongs to the class A1(U) if f and its derivative f ′ admit continuous extensions to the

closure U . We will use the same notations, f and f ′, for the extensions. We endow A1(U)

with the seminorm

‖f‖1 = maxz∈U |f ′(z)|. (2.4.1)

If f ∈ A1(U), f |U is a homeomorphism onto its image and f ′ does not vanish on U ,

we say that f |U is a diffeomorphism onto its image. If U is a bounded connected open

set then ‖ · ‖1 is a Banach norm in the subspace of functions vanishing at a given point

z ∈ U .

Definition 2.4.2. A map f ∈ A1(U) is called a puzzle map if

• ∪Ui is a countable union of quasi-disks Ui, i ≥ 0, called puzzle pieces, with pairwise

disjoint closures, and U0 3 0;


• for i > 0, f is a diffeomorphism of Ui onto some Uj;

• there exists a sequence ni such that fni |Ui is a diffeomorphism onto U0;

• 0 is a critical point of f and f ′ does not vanish on ∂U0;

• f |U0 is a degree d covering map onto its image;

• infi distC\S(∂Ui, f(U0)) > 0;

• infi 6=j distC\S(Ui, Uj) > 0;

where S = C \ ∪∂Uj.

Proposition 2.4.12. Suppose that f is a Yoccoz map with bounded geometry, and I0 ⊃

I1 ⊃ I2 ⊃ . . . is its principal nest. Then if n is sufficiently big, and Ai is the first

landing map associated to In = A0, there exists a collection of open, real-symmetric

quasi-disks Ui in C with Ui ∩ R = Ai, such that f : ∪Ui → C is a puzzle map.

Suppose that Ui is a collection of open, real-symmetric domains in C. We say that

a map f : ∪Ui → C is a pseudo-puzzle map if

1. f : U0 → f(U0) is a degree d covering map,

2. for i 6= 0, there exists l such that f l : Ui → f(U0) is a univalent covering map,

3. For all i either Ui ⊂ f(U0) or Ui is disjoint from f(U0).

Notice that we allow the domains Ui to intersect one another.

Suppose that In0 is a central real puzzle piece of the principal nest for a Yoccoz map

f , and let Inj be the real puzzle associated to In0 .

Lemma 2.4.13. Suppose that f is a Yoccoz map with bounded geometry. Then if In is

a sufficiently deep real puzzle piece in the principal nest and let Inj be the real puzzle

associated to In as above. Then there exists a pseudo-puzzle map f : ∪Ui → C such that

Ui ∩ R = Ini .


Proof. Since ω(0) is a minimal set for f , each point x ∈ ω(0) is in the domain of the

real puzzle associated to In. By compactness, there exists a finite covering of ω(0) by

real puzzle pieces associated to the first landing map to In: In, In1 , . . . , Ink . Let us first

consider a component Inj with j 6= 0 so that there exists l > 0 such that f l : Inj → In is a

diffeomorphism. Provided that n is sufficiently big, there exists a domain Dj ⊂ Dπ/2(Inj )

such that f l : (Dj) → Dπ/2(In) is a diffeomorphism. Now consider the critical puzzle

piece In, and let s be such that f s : In → In−1 is a degree d covering map. The map f s−1

maps f(In) diffeomorphically onto In−1. Hence, there is a region D′ 3 f(0) contained in

Dπ/2(f(In)) that is mapped diffeormorphically onto Dπ/2(In−1). Since In1 is well inside

f(In), D1 is well inside D′, and as a consequence D0 is well inside f s(D0).

Now, we need to deal with those puzzle pieces that do not intersect the post-critical

set. Consider an real puzzle piece I that does not intersect the post-critical set, and let l

be such that f l : I → In is a diffeomorphism. Let Di be the pullback of D0 corresponding

to I. We use the following improvement on the Schwarz Lemma due to de Faria and de

Melo [dFdM]: there exists ε > 0 such that the spoiling factor in the Schwarz Lemma

in the angles of the Poincare domains when going from gj+1(Di) to gj(Di) is at most

(1 +K|gj(I)|1+ε) for some universal constant K. Since the orbit of I up to its first entry

in D0 is disjoint and the maximum size of the intervals in the orbit is small when k is

large, we get thats∏j=1

(1 +K|gj(I)|1+ε)

can be assumed to be as close to 1 as we like. Hence Di is well defined, and inside a disk

with angle close to π/2.

2.4.8.1 Construction of a smooth puzzle map

Suppose that we are given a pseudo-puzzle map f : ∪Di → C based on a real puzzle map

f : ∪Ini → I, associated to the first landing map to In (so that Di ∩ R = Ini ).

Choose a neighbourhood V of In−1, a topological disk with analytic boundary. It is


impossible to extend f s|I0 analytically to a domain U0 in such a way that f s : U0 → V

becomes an analytic covering map (if such an extension did exist, its preimage would not

be close to the real line), so instead choose an analytic extension near I0 and then extend

it diffeomorphically so that f s : U0 → V becomes a smooth branched covering map with

critical point 0. Also, choose U0 so close to In that it is well inside V , for each i the

map g : Ui → U0 is an analytic (diffeomorphic) covering map, where Ui is the pullback

of U0 through the first landing map of Ini to In, and the Ui are all properly contained in

Dπ/2(Ini ). Notice that the Ui are not necessarily contained in V when Ij intersects In−1.

Now, fix a topological disk A that properly contains V and Ui whenever Ini intersects

In−1, but has a definite distance from the Ui whenever Ini is not contained in In−1 as in

the definition of a puzzle map. Given i, choose a disk Bi that is properly contained in A,

in case Ini intersects In−1, and that properly contains Ui so that all the Bi are properly

contained in Dπ/2(Ini ). We now smoothly extend the landing map g : ∪Ui → U0 to a

smooth puzzle map g : Bi → A.

Let us remark that g need not coincide with g wherever they are both defined and g

is not analytic; however, they do coincide close to the intervals Ini .

2.4.8.2 Intersecting the smooth puzzle map with a pseudo-puzzle map

Given a unimodal map f , let n be sufficiently big, and let fn(t) : ∪Dni (t) → C be

a smoothly varying family of pseudo-puzzle maps such that Ini = Dni (t) ∩ R, fn(0) is

the pseudo-puzzle map we constructed above and fn(t) agrees with fn(0) on I. For

convenience, we will shift our indices by letting n = 1 so that f1(t) : ∪D1k(t)→ C is our

family of maps. Let g1 : ∪B1i → A1 be a smooth puzzle map so that A1 ∩R = I0 and g1

agrees with the first return map of f1(t) to I1.

By Sard’s Theorem, there exists t close to zero so that the boundaries of the ranges

of f1 and g1 intersect transversally. Therefore, we can find t close to zero such that the

component, U , of A1 ∩D0(t) that contains 0 is a quasi-disk.


The return map to U is analytic since it’s contained in some small Poincare neigh-

bourhood of I0. Let U1 be the pullback of U under g1 (which agrees with the pullback

under f1 close to the real line), and consider the first landing map to U1 under g1. La-

bel the components by U1i so that U1

i ∩ R = I1i . Since U1

i ⊂ Bi all the components of

the domain are disjoint. Indeed, it is easy to ensure that the puzzle map we have just

constructed satisfies the following conditions. Let Q = ∪∂Uj, and let S = C \Q, then

infi 6=j

distC\S(Ui, Uj) > 0, (2.4.2)

and

infi

distC\S(∂Ui, U) > 0. (2.4.3)

In the construction above we ensured that that mod(U \ U1) > 0. Pulling this space

back by univalent branches of the landing map to U1 we have that distC\S(U1i , U) > 0

whenever Ui ⊂ f(U0). Moreover, the first fact follows from our construction since each

Bi is contained in Dπ/2(Ini ), and similarly, we can ensure that distC\S(Ui, U) is uniformly

bounded away from 0 in i whenever Ui ∩ U = ∅.

Thus we get a true puzzle map:

P : ∪Ui → C

By considering the components of the domain of a first return map rather than a

landing map, we can obtain through a simialar argument:

Theorem 2.4.14. Suppose that f is a real-analytic unimodal map that is at most finitely

renormalizable with minimal post-critical set and recurrent critical point. Then f pos-

sesses a generalized polynomial-like generalized renormalization.

To see that the puzzle we have constructed moves holomorphically, we will use the

following general lemma.


Lemma 2.4.15 ([ALM]). The map fv = f (id + v) has an invariant Cantor set Qv ⊂

Z, Q0 ≡ Q, which moves holomorphically over some neighbourhood V ⊂ Λf of 0.

This lemma together with equations (2.4.2), (2.4.3) and the fact that only finitely

many puzzle pieces lie outside a holomorphically moving Markov family containing Q,

imply that the puzzle is persistent.

Lemma 2.4.16 (Lemma 5.6 [ALM]). The puzzle constructed above moves holomorphi-

cally over a neighbourhood f ∈ V ⊂ Ua.

Chapter 3

Hybrid Lamination

3.1 Hybrid classes

One major step in this work is to show that all but countably maps in a non-trivial family

of unimodal maps possess a neighbourhood endowed with a codimension-one lamination

L. In case f is non-regular, we will show that the leaf of the lamination through f is

given by the topological equivalence class of f and we define this to be the real hybrid

class of f . Since the topological equivalence classes of regular maps are open sets, there

is no hope for them to yield a lamination. In order to give a global characterization of the

leaves of L we need to introduce a fixed, albeit arbitrary, refinement of the topological

classes of regular maps. This was carried out in [AM3].

Suppose that f is a regular map, and let A be the set of attracting periodic orbits of f

and let B = x ∈ I|fn(x)→ A. If f is regular, there exists a minimal m ≥ 0 such that

fm(0) belongs to a periodic connected component of B. If f is a quasi-polynomial map,

then m = 0 and we can define the real hybrid class of f to be the set of all regular maps

topologically equivalent to f with the property that the multiplier of the periodic orbit

that attracts 0 is the same for both maps. We will call the case that m = 0 essential.

In order to treat the non-essential case, we consider a subset of the topological class

53

Chapter 3. Hybrid Lamination 54

of f given by more restrictive topological equivalences.

If f is a regular map of non-essential type we define the real hybrid class of f to be

the set of all regular maps g such that there exists an f -admissible topological conjugacy

between f and g.

The following remark will be important in some of the pullback arguments we will

use later (see [AM3] Remark A.3). If g1, g2 ∈ V∩Ua are regular maps in the same hybrid

class. Suppose |In|/|In−1| is sufficiently small Let A0 = In, and suppose that ∪Aj is

the associated real puzzle with A1 = [l, r]. Let A denote the collection of puzzle pieces

contained in [−1, r]. By the Schwarz Lemma, g1 and g2 are of non-essential type if and

only if for all m sufficiently big,

h−1g1

(gm1 (0)), h−1g2

(gm2 (0)) /∈ [−1, r] \ A.

Moreover, the definition of hybrid class implies that h−1g1

(gm1 (0)) = h−1g2

(gm2 (0)).

We let HRf ≡ HR

f,a ⊂ Ua denote the real hybrid class of f .

If f is preperiodic, we let Cf denote the set of maps topologically equivalent to f

and with the same relation on the critical orbit. If f is hyperbolic, Cf will denote

the hybrid class of f . We say that a preperiodic or hyperbolic complex map g has

special combinatorics with respect to V , a complex neighbourhood of g, if the connected

component of g ∈ V ∩ Cg contains a real map.

For hyperbolic and preperiodic maps the arguments of [ALM] and [AM3] go through

without change, and the following proposition is a statement of the first part of Theorem

A in these cases.

Proposition 3.1.1. The hybrid class of a hyperbolic or preperiodic map is a codimension-

one embedded real analytic Banach submanifold in Ua.


3.2 Infinitesimal theory

3.2.1 A variational formula

Let Sn denote the iteration operator f 7→ fn as a map acting between spaces of analytic

functions. Linearizing (f + εv)n, we obtain by induction the following formula for the

differential of Sn.

vn ≡ DSn(f)v = Dfn−1 fn−1∑k=0

v fk

Dfk f= Dfn

n−1∑k=0

(fk)∗(v

f ′). (3.2.1)

Note that if ft is a one-parameter family of analytic maps such that

d

dtft

∣∣∣∣∣t=0

= v,

then

d

dtfnt

∣∣∣∣∣t=0

= vn. (3.2.2)

Applying (2.2.1) to the iterates of f ∗ and summing up, we see that if α is equivariant

with respect to (f, v) on ∪n−1k=0f

k(X), then it is equivariant with respect to (fn, vn) on X:

(fn)∗α− α =n−1∑k=0

(fk)∗(f ∗α− α) =n−1∑k=0

(fk)∗(v

f ′) =

vn

Dfn(3.2.3)

Rewriting this equation, we have

α = (fn)∗α− vn

Dfn, (3.2.4)

and now using (3.2.1), we see that if α is a bounded vector field on orbf (x) and Dfk(x)→


∞ then we have

α(x) = −∞∑j=0

v f j(x)

Df j+1(x).

Finally note that if β is obtained by n consecutive lifts of α by (f, v), then β is the

lift of α by (fn, vn).

3.2.2 Macroscopic pullback argument

In this section f : U f → C and f : U f → C will be assumed to be puzzle maps, and h

will denote a homeomorphism such that h(U f ) = U f , equivariant with respect to f on

∂U f . If h(f(0)) = f(0). Since f and f are univalent on the non-central components of

their domains and branched coverings on their central components, we can lift h to the

map h1, which coincides with h on C \U f and such that h f = f h1 on U f . Since the

first landing maps to the central domain of a generalized polynomial-like map is a puzzle

map, the results below hold in that setting as well.

Definition 3.2.1. We say that a homeomorphism h : C→ C is a combinatorial equiva-

lence between f and f if it is equivariant on ∂U f and the lift h1 of h is homotopic to h

rel ∂U f ∪ orbf (0).

This Sullivan’s pullback argument, but for puzzle maps.

Theorem 3.2.1 ([ALM]). Let us consider two puzzle maps f and f with all periodic

orbits hyperbolic. Let h be a qc combinatorial equivalence between f and f . Then there

is a qc homeomorphism H : C → C such that H f = f H on U f , H = h on C \ U f ,

and Dil(H) ≤ Dil(h). If there are no invariant line fields on K(f) or if f and f are

hyperbolic maps in the same hybrid class, then Dil(H) ≤ Dil(h|C \ U f ).

One consequence of the Pullback argument is that conjugacies between preperiodic

or hyperbolic maps close to Yoccoz maps respect the puzzle structure:


Lemma 3.2.2 ([ALM]). There exists a constant L > 0 with the following property. Let

f ∈ Ua be a Yoccoz map, and let Pf be a puzzle which persists in an ε-neighbourhood of

f ∈ A. Let V be an ε/2-neighbourhood of f . If g ∈ Ua ∩ V is preperiodic or hyperbolic

and g belongs to the same connected component of Cg∩V, then there is a normalized L-qc

homeomorphism h : C→ C equivariant with respect to g and g on U g.

3.2.3 Infinitesimal Pullback Argument

Given a puzzle map, f , the Infinitesimal Pullback Argument allows us to extend qc vector

fields that are defined and equivariant on certain parts of the domain of f , for example

on the post-critical set of f or on the boundary of the domain of f , to the entire domain.

The next lemma provides one step of the infinitesimal pullback argument.

Lemma 3.2.3. Let Ω 3 0 be a quasi-disk. Consider a map f ∈ A1(Ω) whose derivative

does not vanish on Ω \ 0. Assume that f : Ω → f(Ω) is either a diffeomorphism or

a degree d branched covering ramified at 0. Let v ∈ B(U). Let α and β be qc vector

fields on C such that β|∂Ω is the lift of α by (f, v). Moreover, if f is a degree d branched

covering, we assume that v(0) = α(f(0)), and v(i)(0) = 0 for 1 ≤ i ≤ d − 2 (notice that

this restriction is irrelevant in the case when d = 2). Then there exists a qc vector field

γ such that γ|Ω is the lift of α by (f, v), γ|C \ Ω = β, and

‖∂γ‖∞ ≤ max ‖∂α‖∞, ‖∂β‖∞.

Proof. Define a continuous vector field γ on C\0 by letting γ = β on C\Ω and letting

γ = (α f − v)/f ′ on Ω \ 0.

If f is a diffeomorphism then γ clearly extends to 0.

Now, assume that f is a degree d branched covering. Since the modulus of continuity


of qc vector fields is φ(x) = −x ln(x), for z near 0 we have:

|α(f(z))− α(f(0))| = O(φ(|f(z)− f(0)|)) = O(φ(|z|d)) = O(−d|z|d ln(|z|))

Since v(0) = α(f(0)) and f ′ has a zero of degree d− 1 at 0, we have that near 0

γ(z) =α f(z)− v(z)

f ′(z)

=α f(z)− α f(0) + α f(0)− v(z)

f ′(z)

=v(0)− v(z)

f ′(z)+α f(z)− α f(0)

f ′(z)

=v(0)− v(z)

f ′(z)+O(

−zdln(z)

zd−1)

Hence,

γ(z) =v(0)− v(z)

f ′(z)+O(φ(|z|).

Thus γ has a continuous extension to 0 since v(i)(0) vanishes for 1 ≤ i ≤ d− 2 .

γ is quasi-conformal on C \ (∂Ω ∪ 0) (because α is qc), and since quasi-arcs and

isolated points are qc removable, γ is quasi-conformal on all of the complex plane.

Since lifts preserve the norm of qc vector fields, we have that ‖∂γ‖∞ = ‖∂α‖∞ on

Ω, while we have that ‖∂γ‖∞ = ‖∂β‖∞ on C \ Ω. Since quasi-arcs are removable, the

desired estimate follows.

With this lemma established we can proceed exactly as in [ALM] to prove:

Theorem 3.2.4 (Infinitesimal Pullback Argument). Let f : U → C be a puzzle map

whose critical point does not escape U , and let v be a tangent vector field at f . Assume

that there exists a normalized qc vector field β on C which is equivariant on ∂U ∪ orb(0).

Then there exists an equivariant on U qc vector field α with ‖∂α‖∞ ≤ ‖∂β‖∞ which

coincides with β on C \ U . Furthermore, if there are no invariant line fields on K(f),

then ‖∂α‖∞ ≤ ‖∂β|C \ U‖∞.


We let Lg denote the linear map that associates with any tangent vector v ∈ Tg the

unique qc vector field α on the post critical set such that v = αf−f ′α and v(0) = α(c1).

We will include the proof of the following theorem for the sake of completeness.

Theorem 3.2.5 (Key Estimate, [ALM]). Let f ∈ Ua be either a Yoccoz map or a hy-

perbolic map. There exists a neighbourhood V of f in Aa and a constant C > 0 such

that, for any g with special combinatorics with respect to V, the operator norm of Lg is

bounded by C.

Proof. First, suppose that f is a Yoccoz map, and consider a persistent puzzle for f .

Take an ε > 0 such that the puzzle persists in an ε-neighbourhood of f , and let Hg

be the associated holomorphic motion. Let V be an ε/2-neighbourhood of f and let

C = 2/ε. Given g ∈ V with special combinatorics and v ∈ Tg with ‖v‖a = ‖v‖∞ = 1, let

hλ = Hg+λv H−1g , λ ∈ Dε/2. Let

β0 =d

dλhλ

∣∣∣∣∣λ=0

.

Notice that β0 is equivariant on ∂U g with respect to (g, v). By the Bers-Royden Theorem,

2.2.8, µhλ is a holomorphic function on Dε/2 with values in the unit ball of L∞(C). By

Lemma 2.2.12,

∂β0 =d

dλµhλ

∣∣∣∣∣λ=0

.

Since µh0 = 0, we can apply the Schwarz Lemma to get that ‖∂β0‖∞ ≤ 2/ε.

Assume that g is preperiodic. Then it is well known that g has no invariant line fields

on its filled Julia set.

By Lemma 3.2.2, the special combinatorics assumption implies that g|U g is qc con-

jugate to g|U g for some real map g ∈ V ; in particular, the critical orbit does not escape

U g. By Theorem 3.2.4, there exists a qc vector field α equivariant on U g, coinciding with

β0 on C \ U , and such that ‖∂α‖∞ ≤ C.


In the hyperbolic case we proceed exactly as in [ALM].

Since real maps do not possess invariant line fields, the Key Estimate and the In-

finitesimal Pullback Argument give us the following extension theorem for qc vector

fields.

Lemma 3.2.6 ([ALM]). Given (ψ, γ), there exists an L > 0 with the following property.

Let f be a geometric puzzle map. Let v ∈ Υ(Dγ(J1)) be a holomorphic vector field such

that there is a qc vector field α on Of with α(c1) = 0, satisfying v = f ∗α−α. Then α has

a normalized qc extension which satisfies v = f ∗α− α on U and vanishes on ∂(U ∩ R).

In the case of generalized polynomial-like maps we have the following estimate which

will be used to show that a certain vector field is non-degenerate.

Lemma 3.2.7. Let f ∈ Bnor(U) be such that f has a generalized polynomial-like extension

compatible with U . Let V ⊂ U be a neighbourhood of K(f) so that V ′ = f−1(V ) b V .

Then there exists C such that for any g close to f so that 0 does not escape from V and

for v = α g − g′α ∈ Ev(g), we have

‖α‖sphC\V ≤ C‖v‖V and C−1‖v‖V ′ ≤ ‖v‖V

Proof. Let γ = ∂V . A branch of f−1 is a covering map from V to f−1(V ) and the inclusion

of f−1(V ) into V is a contraction in the hyperbolic metric. So f−1 is a contraction on

V in the hyperbolic metric on each component of f−1(V ). In a compact neighbourhood

of K(f) all metrics are equivalent. So some iterate of f will eventually overcome the

constant giving the equivalence, and we get that there exists λ−1 < 1 and N , depending

only on mod(V \ V ′), such that |D(f−N)|γ < λ−1.

Let ‖v‖V = ε and ‖α‖C\V = M . It is obvious that ‖α‖C\V ≤ ‖α‖C\f−N (V ). By the

maximum principle, ‖α‖γ ≤ ‖α‖f−N (γ). Recall that α and v satisfy vf ′

= f ∗(α)− α on U .


Applying the variational formula (3.2.4), we get

α = (fN)∗(α)−N−1∑k=0

(fk)∗(v

f ′)

So that

‖α‖f−N (γ) ≤ ‖(fN)∗(α)−N−1∑k=0

(fk)∗(v

f ′)‖f−N (γ)

≤ ‖(fN)∗(α)‖f−N (γ) + ‖N−1∑k=0

(fk)∗(v

f ′)‖f−N (γ)

≤ ‖α fN

DfN‖f−N (γ) + ‖

N−1∑k=0

(v fk

(Df fk)(Dfk))‖f−N (γ)

≤ M

λ+ε

λ‖N−1∑k=0

(DfN

(Df fk)(Dfk))‖f−N (γ) (here we use the maximum principle)

=M

λ+ε

λ‖N−1∑k=0

(DfN

Dfk+1)‖f−N (γ)

=M + Aε

λ.

So we have that

M ≤ Aε

λ− 1.

Which gives us the estimate

‖α‖C\V ≤ C‖v‖V .

Now, since v = α f − f ′α, we have

‖v‖V ′ ≤ ‖α f‖V ′ + ‖f ′α‖V ′ ≤ C‖v‖V + C|f ′|V ′ |v|V = (C + CK)‖v‖V

If g is sufficiently close to f in Bnor(U) with non-escaping critical point, the same proof

can be carried out with g in place of f and the constants do not change by too much

since they depend only on mod (U ′ \ U).


3.3 Hybrid lamination for maps with sufficiently big

geometry

Suppose that fλ is a non-trivial one-parameter analytic family of unimodal maps with

critical point of fixed degree d. Throughout this section f ≡ f0 ∈ fλ will be a Yoccoz

map with sufficiently big geometry and recurrent critical point. A vector field v ∈ TfAa

is called horizontal if there exists a qc vector field α that is (f, v) equivariant on the

critical orbit. We denote this space of vector fields by Tf . It follows from [ALM] that Tf

is the tangent space to the hybrid class of f . In this section, we will prove:

Theorem 3.3.1. Let f ∈ Ua be a Yoccoz map with sufficiently big geometry and recurrent

critical point. The real hybrid class of f is a codimension-one real analytic Banach

submanifold of Ua.

3.3.1 Splitting of the tangent space

Notice that the tangent vector to the curve t 7→ ftv at t = 0 is given by the vector field that

equals v(z)f ′(z) on U1 and vanishes elsewhere. This vector field is tangent to the hybrid

class of f if there exists a qc vector field α on the orbit of the critical value c1 such that

α(c1) = 0 and v = f ∗α−α. We say that a vector field w is formally transverse at f if there

is no quasi-conformal vector field α on C such that for z ∈ orbf (0), w(z) = f ∗α(z)−α(z).

We let V trf denote the space of formally transverse vector fields.

Suppose that |In|/|In−1| is very small and let A0 = In. Let Aj be the real puzzle

associated to A0 labelled so that the critical value is in A1.

Let us consider an affine map Q : A1 → I, and let

vn(z) = (1− z2)(1− e−2n) +2

n(e−n(1+z) + e−n(1−z) − e−2n − 1),

and let vn ∈ Υ1 be such that vn|Dγ(A1) = Q∗vnε1/8. Notice that ‖vn‖ < ε1.


Lemma 3.3.2 ([ALM], [AM3]). Let vm be defined as above. If |In|/|In−1| is sufficiently

small then for m sufficiently big, vm is formally transverse at f .

Proof. Let us briefly review the proof of this statement. It is clear that each vn ∈ Υ(D),

‖vn‖1 < 6 and vn → 1− z2 pointwise in D. Hence ‖vn(z)‖Dhyp → |1− z2|/(1− |z|2) > 1.

Let n be big and consider the restriction w of vn|Dγ(I), normalized so that ‖w‖1 = 1.

Rescaling Dγ(I) to Dγ(J1) we obtain a vector field v in Dγ(J1) such that ‖v‖1 = 1 and

‖v(c1)‖U1hyp > 1/7 since both the hyperbolic norm and the ‖·‖1 norm are scaling invariant.

Suppose now that there exists a vector field α on C that satisfies v = f ∗α− α on U .

Suppose that n > 1 is minimal such that cn ∈ U1. Since n− 2 is the first landing time of

c1 in f−1(U1), there exists a domain W ⊂ U1 containing c1 that is univalently mapped

onto U0 by fn−2. Since α is equivariant with respect to (f, v),

(fn−2)∗α− α =n−3∑k=0

(fk)∗v.

Using the fact that fn−2 : W → U0 is a hyperbolic isometry and the Schwarz Lemma,

we see that

‖α(cn−1)‖U0hyp ≥ ‖v(c1)‖U1

hyp.

Let q = cn−1. Then [−q, q] ⊂ V ≡ f−1(U1). Since mod(U0 \ V ) > k/d, there exists

a constant depending only on k which bounds dist(q, ∂J0)/diam J0 from below. Since

‖α(q)‖U0hyp > 1/7 and U0 ⊃ Dφ(J0), there is a constant κ depending only on φ and k such

that |α(q)| > κdist(q, ∂J0).

By Lemma 3.2.6 there exists L > 0 depending only on ψ and γ such that α has a

normalized L‖v‖1−qc extension that is equivariant on U and vanishes on ∂(U ∩R). Let

T = f−1J1 ∩ R. Corollary 2.2.14 implies that there exists C depending on L such that

|α(z)| < C|T | for every z ∈ T . If k is big enough then |T | ≤ κC−1dist(T, ∂J). Now, on

the one hand we have

|α(z)| ≤ C|T |κdist(T, ∂J0),


and on the other hand we have

|α(q)| > κdist(q, J0).

This yields a contradiction, and so no such α exists.

From this we know that v must be formally transverse because any qc vector field α

on C such that for z ∈ orbf (0), v(z) = f ∗α(z) − α(z) extends to an L-qc vector field

that is equivariant on U .

We have now established that the transverse direction exists.

Corollary 3.3.3 ([ALM]). The set of vector fields v ∈ Υ which are not formally trans-

verse at f forms a closed subspace of Υ,

This immediately implies:

Proposition 3.3.4. Suppose that f is a Yoccoz map with sufficiently big geometry and

recurrent critical point. Then

TfAa = Tf ⊕ V trf .

Proposition 3.3.5 ([ALM]). Tf is a codimension-one subspace of TfAa.

Proof. Since hyperbolic maps are dense in our family we can approximate any Yoccoz map

f with sufficiently big geometry by hyperbolic maps. Let fn be a sequence of hyperbolic

maps such that fn → f . For each fn, Tfn is codimension-one, so by Lemma 2.2.16 either

lim supTfn is codimension-one or is the entire space. Lemma 8.3 of [ALM] tells us that

if fn → f is a sequence of maps with special combinatorics then lim supTfn ⊂ Tf , and

so since Tf does not contain the transverse vector field it must be a codimension-one

subspace of TAa.


3.3.2 Hybrid lamination

In this subsection, we will complete the proof of Theorem A for Yoccoz maps f as above.

Let Π1 : TAa → Tf and Π2 : TAa → C be the linear projections along Tf and v so that

w = Π1(w) + Π2(w)v. By Lemma 2.2.15, there exists a constant C such that for any

w ∈ TAa \ K, Π2(w) ≤ C‖Π1(w)‖a. Let g ∈ Aa. We say that γ :W → Aa is a g-graphic

(over W) if W is a neighbourhood of 0 in Tf and if γ(0) = g and Π1 γ − id = Π1(g).

We say that a g-graphic is C-Lipschitz if ‖Π2 Dγ‖ ≤ C. We say that a g-graphic

is contained in some set X if γ(W ) ⊂ X, and that a set X is a definite g-graphic if

there is a g-graphic over Λ onto X. A g1-graphic γ1 over W1 and a g2-graphic γ2 over

W2 where W1 ∩W2 is simply connected are said to satisfy the continuation property if

γ1|W1 ∩W2 = γ2|W1 ∩W2.

Fix a puzzle for f and let v denote a transverse vector field. Let V be a neighbourhood

of f as in Lemma 3.2.2 and the the Key Estimate, and let K be an open cone such that

for any g ∈ V with special combinatorics with respect to V , Tg is transverse to K (the

existence of such a cone is ensured by Corollary 8.4 of [ALM]). It follows immediately

that if g ∈ Σ has special combinatorics and γ :W → Cg, W ⊂ Λ is a g-graphic, then γ is

C-Lipschitz. Lemma 8.7 of [ALM] gives us that if g ∈ Σ has special combinatorics, then

Cg contains a definite g-graphic.

Let γg be the definite g-graphic contained in Cg, and let ∆g = γg(Λ) ⊂ Cg. By

the continuation property, if g1, g2 ∈ Σ have special combinatorics, and are different,

then ∆g1 ∩∆g2 = ∅. Since hyperbolic maps are dense in Σ, we can apply the Extension

Lemma for holomorphic motions to see that there is a unique extension of the lamination

∆g whose leaves pass through each point of Σ. Given any g ∈ Σ not necessarily with

special combinatorics, we let ∆g be the leaf of the extended lamination; it is still a

definite C-Lipschitz g-graphic. It follows that the slices of ∆g by Ua form a lamination

with codimension-one real analytic leaves. Moreover, by the First Compactness Lemma

maps in the same leaf are L-qc conjugate, and so the leaves coincide with the hybrid


classes.

3.4 Hybrid lamination for maps with minimal post-

critical set

In this section we will prove:

Theorem 3.4.1. Let f ∈ Ua be a Yoccoz map with minimal post-critical set. The real

hybrid class of f is a codimension-one real analytic Banach submanifold of Ua.

3.4.1 Splitting of the tangent space for generalized polynomial-

like maps

Suppose that f : U → U ′ is a unimodal generalized polynomial-like map: U = U0 ∪U1 ∪

· · · ∪ Un.

Proposition 3.4.2. Suppose that f : U → U ′ is a generalized polynomial-like map. Then

TBnor(U) = Eh(f)⊕ Ev(f).

Proof. Suppose that v is a vector field in the tangent space to a generalized polynomial-

like germ, and let f : U → U ′ be a representative of that germ so that v is well defined

in a neighbourhood of U . Let V be union of the pullbacks of the connected components

of U by the branches of f . Let w be a smooth vector field defined on a neighbourhood

of C \ V such that

v(z) = w f(z)− f ′(z)w(z) for z ∈ Dom(w) ∩ V.

To see that such a w exists, define it in a neighbourhood of ∂U , then define it in a

neighbourhood of ∂V so that it satisfies (2.2.1) there and then extend it smoothly to


C \ V .

Now, using (2.2.1), letting

w(z) =w f(z)− v(z)

f ′(z)

we can extend w to a smooth vector field on C \K(f) satisfying (2.2.1) in V \K(f).

Consider the Beltrami differential µ = ∂w in C \K(f) extended by 0 to the filled Julia

set K(f). Since v is holomorphic, an easy computation taking ∂ of both sides of 2.2.1,

implies that µ is f -invariant on V .

∂v = ∂(w f − f ′w), which implies that

0 = (∂w f)f ′ − f ′∂w, and so

∂w = ∂w f, hence

µ = µ f

Thus µ has bounded L∞-norm equal to its L∞-norm on C \ V . So we can solve the

∂-problem, ∂u = µ, where u(z)/dz is a vector field on C with locally square integrable

distributional partial derivatives, ∂u and ∂u, that vanishes at ∞. The horizontal part

vh of v is given by vh = u f − f ′u on U . To see that it is horizontal, notice that it is

holomorphic since µ is f-invariant, it can be normalized by adding a degree d polynomial

to u, and µ vanishes on K(f).

Let α = w − u. Since ∂α = 0, α is holomorphic, and because both w and u vanish

at ∞ so does α. Moreover v − vh = α f − f ′α on U . Hence (v − vh)/dz is a vertical

vector field.

It remains to prove the uniqueness of the splitting. To this end suppose that there

exists a vector field v(z)/dz ∈ Eh(f) ∩ Ev(f). Then near the filled Julia set we may

express v = w f − f ′w where w ∈ F(f) and v = α f − f ′α where α is holomorphic on

C \K(f) and vanishes at ∞.

Let u(z)/dz = (w(a) − α(z))/dz on U \K(f). Since u is f -invariant (f ∗(u) = u), it


is bounded with respect to the hyperbolic norm on each of the components of U. Hence

|u(z)| → 0 as z → J(f). So u admits a continuous extension to U vanishing on the Julia

set.

Thus the vector fields w(z)/dz and α(z)/dz match on the Julia set. Consequently, the

vector field β(z)/dz equal to w(z)/dz on K(f), and α(z)/dz on C \K(f) is continuous

on the whole sphere. It follows from Lemma 10.4 in [L7], a “gluing” theorem for maps

h with locally square integrable distributional derivatives ∂h and ∂h, that β(z)/dz has

distributional derivatives of class L2 and that ∂β = 0. Applying Weyl’s Lemma, β(z)/dz

is holomorphic on the whole sphere. Since it vanishes at ∞, where it is equal to α,

β(z)/dz is linear.

Thus v(z) = af(z) + b − f ′(z)(az + b) where f(z) = c + zd + c1zd+1 + c2z

d+2 + . . . ,

and v is normalized so that v′(0) = v′′(0) = · · · = v(d)(0) = 0. Computing the derivatives

of v, by induction we get that

v(d−1)(z) = −f (d)(z)(az + b)− (d− 2)af (d−1)(z)

v(d)(z) = −f (d+1)(z)(az + b)− (d− 1)af (d)(z)

Evaluating these equations at 0, we get

−bf (d)(0)− (d− 2)af (d−1)(0) = −bf ′(d)(0) = 0, which implies that b = 0, and

−(d− 1)af ′(d)(0) = 0, which implies that a = 0.

Therefore we have that v = 0 and so the splitting is unique.

Lemma 3.4.3 ([Sm]). Let f be a generalized polynomial-like map with non-escaping

critical point.


(1) The vertical space Ev(f) is finite dimensional in any Banach slice compatible with

f . Furthermore dim Ev(f) does not depend on the Banach slice.

(2) Let U be a domain compatible with f . Then there exists ε > 0 so that, for any

generalized polynomial like map g, with non-escaping critical points and ε-close to f

in Bnor(U), we have dim Ev(f) = dim Ev(g).

Lemma 3.4.4 ([Sm]). Let f be a generalized polynomial-like map with a unique critical

point; assume that there are no invariant line fields on K(f) and let U be a domain

compatible with f . Assume that there exists a sequence of generalized polynomial like

maps fn → f so that fn has a super attracting fixed point. Then dim Ev(f) = 1.

So, since hyperbolic maps are dense in fλ, we can apply Lemma 3.4.3 to see that:

Proposition 3.4.5. For any generalized polynomial-like map f , the tangent space of f ,

TBnor(U), splits into Eh(f), which has codimension-one and is the tangent space to Hf ,

and Ev(f).

3.4.2 Tangent space for unimodal maps with minimal critical

orbit

We pullback the structure of the tangent space of generalized polynomial-like maps to

unimodal maps with minimal post-critical set.

Lemma 3.4.6. The generalized renormalization operator is analytic.

Proof. The generalized renormalization operator is a composition of the iteration oper-

ator with the rescaling operator. The rescaling operator is obviously real analytic, and

the iterates of f depend analytically on f .

Lemma 3.4.7. Suppose that f is a real-analytic unimodal map with minimal post-critical

set and recurrent critical point that is non-renormalizable. Then for any generalized


polynomial-like generalized renormalization of f , R : Ua → Bnor(U) (U = U0∪ U1∪ · · ·∪

Un) such that the image of DR : TUf → TBnor(U) is dense.

Proof. Recall that Sn : f 7→ fn. The derivative of the generalized renormalization

operator is the composition of DSn with the derivative of a rescaling map. The image

of the rescaling map is clearly dense in its range so all that remains to be shown is that

the image of Sn is dense onto its range. To see this recall the formula

DSnv = Dfnn−1∑k=0

(fk)∗(v

f ′).

Let w be a vector field defined in U0 ∪ U1 ∪ · · · ∪ Un. We now define a vector field v so

that DS(f)v = w. First, set v = 0 on

∪ni=0 ∪nik=1 f

k(Ui)

Notice that the sets being considered here are the forward images of the components of

domain of Rf under the original map f , until they land on the range of Rf . When f is

non-renormalizable, the closures of these sets are pairwise disjoint.

Now using the formula above with w on the left hand side, extend v to ∪ni=0Ui. Notice

that all the terms in the sum vanish except the one corresponding to k = 0, and we have

that

v(z) =w(z)

Dfn−1(f(z))

v is a vector field vanishing outside U0 such that DSn(f)v = w. To complete the proof,

we use Runge’s theorem to approximate v by an even R-symmetric polynomial vector

field.

Lemma 3.4.8. In the setting of the previous theorem, the map DR : TUa → Bnor(U) is

transversally non-singular.

Proof. Since the image of the derivative of the generalized renormalization operator is


dense, its image contains vectors transverse to HRf .

This immediately implies:

Proposition 3.4.9. The tangent space to a map f with minimal post-critical set splits

as

TfUa = Eh(f)⊕ Ev(f).

3.4.3 Infinitely renormalizable maps

In the case of infinitely renormalizable maps we have to deal with the possibility that the

domains of the landing map to U are not disjoint. This situation occurs when the maps

are satellite renormalizable (when in fact the closures of the little Julia sets intersect).

We will briefly review the the argument of [ALM], which is used to show that in the case

of satellite renormalization, the image of the renormalization operator is dense. This

is the only place in the above argument where the disjointness of the domains was of

importance.

Suppose that g : U → U ′ is a polynomial-like map that arises as a satellite renormal-

ization of an infinitely renormalizable unimodal map f . Suppose that fp(U) = U ′ and

let Ui = f i(U), 0 ≤ i ≤ p. If U0 ∩Un intersect, we may assume that their intersection is

contained in a small neighbourhood V of a repelling periodic point q of period n.

Let w ∈ BRU be an arbitrary even polynomial vector field (such vector fields are dense

in BRU). There exists a holomorphic vector field vlin in fn(V ) such that

w(z) = Dfn−1(fn+1(z))vlin(fn(z)) +Df 2n−1(f(z)), z ∈ V.

Let vn be an even polynomial vector field which is close to vlin in fn(V ), and let v0 be

an even polynomial vector field close to

w(z)−Dfn−1(fn+1(z))vn(fn(z))

Df 2n−1(f(z))


in a neighbourhood of U0 ∪ V . The importance of these vector fields is that they are

compatible under iterations.

Let U symn = U ∪−U , and V sym = V ∪−V , and let K be a 0 symmetric neighbourhood

of U0 ∩Usym

n . Using a partition of unity, we can construct a smooth vector field vsm that

interpolates between these vector fields such that vsm = v0 on U0, vsm = vn on U symn \K,

and vsm is C1 close to vn on K.

Let α be a normalized qc vector field such that ∂α|U0∪U symn = ∂vsm and ∂α|C\ (U0∪

U symn ) = 0. The vector field vsm − α ∈ B0

U0∪Usymn

interpolates between v0 and vn.

By the Mergelyan Approximation Theorem, we have an an even polynomial vector

field v that is close to v0 on U0, vn on Un and to 0 on the Ui, 1 ≤ i < n with the property

that DR(f)v is close to w in B(U).

3.4.4 Lamination near maps with minimal post-critical set

Since we may associate a persistent puzzle map to a non-renormalizable map f with

minimal post-critical set, the arguments presented in Section 3.3 go through without

change for these maps once the splitting of the tangent space, TUa, into the vertical

direction and the tangent space to the hybrid class of f is established.

Chapter 4

Regular or Stochastic Dynamics

4.1 Parameter partition

In this section we prove the following theorem giving us priori bounds in the principal

nest of parapuzzle pieces.

Theorem 4.1.1. Suppose that f is a Yoccoz map with recurrent critical point. There

exists a δ > 0 such that

• mod(V i \ V i+1) > δ,

• mod(V i \ V i+1) > δ,

where V 0 ⊃ V 1 ⊃ V 2 ⊃ . . . , is the principal nest for f , and V1 ⊃ V2 ⊃ V3 ⊃ . . . is the

principal nest of parapuzzle pieces for f such that each V i consists of those maps that are

topologically conjugate to f up to depth i− 1.

We divide the proof of this result into two cases: when the maps has sufficiently big

geometry and when its post-critical set is minimal.

73

Chapter 4. Regular or Stochastic Dynamics 74

4.1.1 Sufficiently big geometry

4.1.1.1 Special family

When we have sufficiently big geometry the infinitesimal transversality of vm in fact gives

us macroscopic transversality.

Lemma 4.1.2 ([AM3]). There exists a constant τ0 > 0 depending only on ε1 and φ, such

that if |In|/|In−1| is sufficiently small the following holds. Let vm be defined as above

and let r > 0 be minimal with f r+1(0) ∈ V1. Then for m sufficiently big, there exists a

domain Θ ⊂ D such that the map θ : Θ→ C given by θ(λ) = f rλvm(0) is a diffeomorphism

onto Dτ0|In|

This, when combined with a generalized renormalization procedure yields:

Proposition 4.1.3 ([AM3]). If |In|/|In−1| is small enough there exists a real-symmetric

vector field v that is transverse to the hybrid class of f and a neighbourhood v ∈ V such

that for any w ∈ V real-symmetric, there exists a domain 0 ∈ Θ ⊂ D, a family of R-maps

R1[λ] : ∪jU1j → V 0, λ ∈ Θ, and a real-symmetric holomorphic motion h over Θ such

that:

(1) For λ ∈ Θ ∩ R, U10 [λ] ∩ R = In+1;

(2) R1[λ] is the first return map from ∪jU1j to U1 under iteration by fλ;

(3) (R1[λ], h) form a full real-symmetric R-family.

And moreover, if additionally w has an analytic extension w : I → I of C1-norm less

than one with w(1) = w(−1) = 0 and λ ∈ Θ ∩ R then:

(4) In+1[λ] ≡ U10 [λ] ∩ R is the component of the first return map to In under iteration

by fλw;

(5) In+1j [λ] ≡ U1

j [λ]∩R are the domains of the return map to In+1 under iteration of the

real analytic extension fλw : I → I.


We also have good control over the geometry of the special family.

Proposition 4.1.4 ([AM3]). In the setting of the previous lemma Dil(h|C \U1

0) < 1 + ε,

and mod(U1[0] \ U10 [0]) > C, where ε→ 0 and C →∞ when |In|/|In−1| → 0.

4.1.1.2 Proof of Theorem 4.1.1 for maps with sufficiently big geometry

By Theorem 4.1.3 we can construct the special family near f . Since, by Proposition 2.3.3

we have a priori bounds for R-maps in the dynamical plane, Proposition 4.1.4 implies

that near f , this family has a priori bounds in the parameter plane. By Section 3.3, f

possesses a neighbourhood that is laminated by the hybrid classes, and the holonomy

map of this lamination is K-qc with small dilatation. So we can transfer the bounds for

the special family to the original family via the holonomy map.

4.1.2 Maps with bounded geometry

Suppose that f0 is a Yoccoz map with bounded geometry in a transverse one-parameter

family fλλ∈Λ and let I0 ⊃ I1 ⊃ . . . be its principal nest of real puzzle pieces. Then, for

n sufficiently big there exists a puzzle map f : ∪Uj → C associated to the domain of the

landing map to In. Now, we can construct the principal nest for the puzzle map by letting

V 0 = f(U0), V 1 = U0, and V i+1 be the component of the domain of the return map to

V i that contains the critical point. We can now consider the puzzle maps fi : ∪V ik → V i

obtained by considering the first landing map to each V i.

Lemma 4.1.5. There exists a constant C > 0 such that if the critical orbit passes through

a puzzle piece V mk that is mapped onto V m such that mod(V m−1 \ V m

k ) > C, then there

exists a level of the principal nest such that mod(V m+n−1 \ V m+n) is so large that f has

sufficiently big geometry.

Proof. We may assume that m = 0. Suppose that ω(0) intersects a puzzle piece V∗ ≡ V 1k

such that mod(V 0\V∗) ≥ α. We may assume that V∗ is a non-central puzzle piece because


if it was central, the lemma would follow immediately. Let l be such that f l(0) is the

first entry of the post critical set into V∗. We let k be such that fk maps V∗ univalently

onto V 1.

Note that mod(V 0 \ V∗) is very large. Let us pull (V 0, V ∗) back along the critical

orbit under f l to obtain a pair (W ′,W ). The map f l : W ′ \W → V 0 \ V∗ is a d-to-1

covering map. Hence mod(W ′ \W ) = 1dmod(V 0 \ V∗) ≥ 1

dα.

Notice now that W ′ is a child of V 0 since it is mapped as a d-to-1 covering onto V 0

by f l and that W is a child of V 1 since it is mapped onto V∗ as a d-to-1 covering by f l

and then mapped univalently onto V 1 by fk. Hence, we have the following nest of puzzle

pieces:

V 0 ⊃ V 1 ⊃ W ′ ⊃ V 2 ⊃ W ⊃ V 3.

So taking α sufficiently big we can ensure that either mod(V 1 \ V 2) or mod(V 2 \ V 3)

is very large.

With this lemma in mind, at each level n of the puzzle, we restrict the domain of the

puzzle map to just those pieces in V n−1 that are so big that if the critical orbit passes

through them we do not have sufficiently big geometry and their pullbacks. Consider the

restriction of the return map to V n, Rnf : Uni → V n, where each Un

i either intersects

the post-critical set or contains one of the finitely many sufficiently big components of

the domain of the first landing map to V n+1 ≡ Un0 . In this way, we obtain a sequence

Rnf : ∪nk=0Unk → V n of generalized polynomial-like maps. The extra care is needed to

ensure that we do not ignore any combinatorial classes.

By Lemma 3.4.6, Rn is an analytic map, and so it is an open map. Hence it maps

neighbourhoods of f0 to neighbourhoods of Rf0. So if V ⊂ Aa is a small neighbourhood

of f0, then RnV is a family of generalized polynomial-like maps such that Rn(V ∩ Λ) is

a family of real generalized polynomial-like maps that is transverse to the topological

class of Rnf0. After sufficiently many generalized renormalizations of f0 we can find a


family of generalized polynomial-like maps, gλ : U1j [λ] → V 0[λ] over a subset of a one

dimensional complex disk V1 ⊂ V such that the family gλλ∈V1 satisfies the conclusion

of Lemma 2.3.4. To see this, consider the image fλ(0) as λ varies over RV . Let Ln denote

the first landing map of the critical value in V n. For n sufficiently big, we can find a

subfamily J ⊂ R(V ∩ Λ) such that as λ varies over J , gλ(0) varies over the preimage

of V n ∩ R under Ln in U11 . Let us choose now a complex disk D ⊂ RV that contains

J . Then the image of the critical value map gλ(0)λ∈D contains an open disk D′ in U11

that contains the critical value. Choosing n even bigger, if necessary, D′ contains the

preimage W of some V n under Ln. Restricting D if necessary we obtain an unfolded

family of generalized polynomial-like maps.

Let us reindex the domains above so that V 0 is the domain V n that we just found.

Then, for the family gλ : ∪U1j [λ]→ V 0[λ]λ∈D, the critical value map is a diagonal to the

tubing given by ∪U1j over D. So we can use it as usual to partition D into parapuzzle

pieces D1j such that the maps in each D1

j have the same combinatorics up to depth 1.

Let V1 be the inverse image of D under the renormalization operator, and let V1j be the

inverse image of D1j so that all the maps in each V1

j have the same combinatorics up to

depth 1. We let V10 = V2. Continuing in this manner we construct the principal nest of

parapuzzle pieces V0 ⊃ V1 ⊃ . . . , for f and partition each Vn by Vn+1j . The fact that

there exists δ such that mod(Vn \ Vn+1) ≥ δ follows from Lemma 2.3.5.

4.2 Slowly recurrent maps

Our goal in this section is to show that almost every parameter is “slowly recurrent.”

To carry out the parameter exclusion argument below, we will need certain intermediate

parapuzzle pieces. The first return map gi+2 : V i+2 → V i+1 may be extended to a map

onto V i. Let V i+2 denote the pullback of V i by this map, and let V i+2 be the associated

parapuzzle piece so that all maps in V i+2 have the same combinatorics up to their first


landing in V i+2. Notice that V i+2 is a good child of V i. The boundary of V i gives us a

tubing over V i+2 with a diagonal given by λ 7→ gi+2[λ](0).

Lemma 4.2.1 ([ALS]). For any δ > 0 and λ > 0 there exists ε = ε(δ, d) and K =

K(d, δ, λ) with the following property. Assume that for some parapuzzle piece Vn,

mod(Vn \ Vn+1) > δ, and mod(V n(c) \ V n+1(c)) > δ, c ∈ Vn+1.

Then for every c ∈ Vn+k+3 such that sn+1(c) ≥ (1 + i)K, i = 0, 1, . . . , k, we have:

(1) mod(V n+i \ V n+i+1) > maxε, (i− 1)λ,

(2) mod(Vn+i \ Vn+i+1) > maxε, (i− 2)λ,

(3) mod(Vn+i+2 \ Vn+i+2) > maxε, (i− 1)λ).

Lemma 4.2.2 ([ALS]). Assume that for some parameter c0

mod(Vn \ Vn+1) > δ, mod(V n \ V n+1) > δ.

Let Znr ⊂ Vn+1 be the set of parameters that are not combinatorially recurrent. Fix some

K that is larger than the K given by the previous lemma, and let Zsr ⊂ Vn+1 be the set

of combinatorially recurrent parameters for which sn+k ≥ (1 + k)K, k ≥ 0.

Then P(Zsr ∪ Znr|Vn+1) > ε(δ, d,K) > 0.

Proof. Let tk = (1 + k)K and let X = Vn+1 \ (Zsr ∪ Znr). For k ≥ 0, 0 ≤ j ≤ tk, let

Xk,j ⊂ X be the set of all c ∈ X such that sn+i ≥ ti, 0 ≤ i < k and sn+k = j. Notice

that

X = t(k,j)Xk,j.


Order the pairs lexicographically, and let

Xk,j = ∪(k′,j′)<(k,j)Xk′,j′ .

Notice that for x ∈ Xk,j we have:

Vn+k+2(c) ∩ Xk,j = ∅,

and

Vn+k+2(c) ∩Xk,j ⊂ Vn+k+2(c).

For c ∈ Vn+k+2(c), we have sn+i(c) = sn+i(c) ≥ tn+i for i < k, while sn+k(c) ≥

sn+k(c) = j, with equality if and only if c ∈ Vn+k+2(c).

So we have that

P(Znr ∪ Zsr|Vn+1) = 1− P(X|Vn+1)

=∏(k,j)

(1− P(Xk,j|Vn+1 \ Xk.j)) ≥∏(k,j)

(1− supc∈Xk,j

P(Xk,j|Vn+k+2(c)).

Thus it is enough to prove and an estimate such as

P(Xk,j|Vn+k+2(c)) ≤ e−(1+k)ε, c ∈ Xk,j

for some ε = ε(δ, d). But this follows from

Vn+k+2(c) ∩Xk,j ⊂ Vn+k+2(c)

and the estimate

mod(Vn+k+2(c) \ Vn+k+2(c)) > (1 + k)ε

of the previous lemma.


4.3 Conclusion

In this section we will quote the results needed to complete the proofs of Theorems B

and C.

An easy counting argument yields:

Lemma 4.3.1. Let fλ be a non-trivial analytic family of unimodal maps. Then at

most countably many parameters are not Kupka-Smale or have periodic or preperiodic

critical point.

The following Lemma is due to Douady:

Lemma 4.3.2 (Lemma 8.2 [AM3]). Let L be a codimension-one complex lamination of

an open subset V of some Banach space, and let γ be an analytic path in V. If γ is not

contained in a leaf of L, then the set of parameters where γ is not transverse to the leaves

of L consists of isolated points.

Hence,

Lemma 4.3.3. Let fλ be a non-trivial one-parameter family of unimodal maps. Then

the set of non-regular Kupka-Smale parameters λ0 such that fλ is not transverse to the

topological class of fλ0 is countable.

From the results of [AL3]

Theorem 4.3.4. Let fλ be a non-trivial analytic family of unimodal maps. Then the

set of infinitely renormalizable parameters has measure zero.

Theorem 4.3.5 ([S]). Let fλ be a non-trivial analytic family of unimodal maps. Then

almost every parameter is regular or has a recurrent critical point.

At this point, we know that the set of finitely renormalizable maps with recurrent

critical point has full measure in the set of non-regular parameters. Let S denote the


set of Yoccoz maps in fλ such that the return times to V n, mod(V n \ V n+1) and

mod(Vn \Vn+1) all grow at least linearly in n. To prove the next theorem, we will exploit

the following density points argument. Let X ⊂ C be a measurable set such that for

almost every x ∈ X there exists a sequence Xn(x) ⊂ C containing x and such that

diam Xn(x) → 0. Assume that any two Xn(x), Xm(y) are either nested or disjoint.

Then lim P(X|Xn(x)) = 1 for almost every x ∈ X.

As a result of Sands’ Theorem, when we restrict Lemma 4.2.2 to the real slice, we

have that there exists ε > 0 such that P(S|Vn ∩ R) > ε. Hence the above density points

argument implies

Theorem 4.3.6. S has full measure in the set of non-renormalizable maps.

The parameters in S satisfy the phase-parameter relations of [AM3] and indeed the

same argument used to prove Theorem B of that paper, that almost every such parameter

has a quasi-quadratic renormalization, applies in the higher degree case to maps in S,

and implies that almost every map in S has a quasi-polynomial renormalization. With

this result in hand, we may apply the statistical arguments of [AM2], which yield:

Theorem 4.3.7. Almost every non-regular parameter in fλ is Collet-Eckmann.

Since, almost every map in S is both Collet-Eckmann and possesses an quasi-polynomial

renormalization, these maps, after renormalization do not possess periodic attractors,

and as a result they possess unique absolutely continuous invariant probability measures

that are ergodic and supported on a finite cycle of intervals. Hence, we can apply the

arguments of [AM2], [AM3] and [AM4] in the higher degree case to prove:

Theorem 4.3.8. Suppose that fλ is a non-trivial analytic family of unimodal maps.

Then almost every fλ ∈ fλ which is non-regular satisfies:

1. The critical point is polynomially recurrent with exponent 1:

lim sup− ln |fnλ (0)|

lnn= 1,


2. The critical orbit is equidistributed with respect to the absolutely continuous invari-

ant measure µ:

lim1

n

n−1∑i=0

φ(f iλ(0)) =

∫φdµ

for any continuous function φ : I → R,

3. The Lyapunov exponent of the critical value, lim1

nln |Dfnλ (fλ(0))|, exists and co-

incides with the Lyapunov exponent of µ.

4. The Lyapunov exponent of any periodic point p contained in suppµ is determined

(via an explicit formula) by combinatorics (more precisely, by the itineraries of p

and of the critical point).

Remark 4.3.1. The argument of Lemma 9.6 of [AM3] applies equally well in this case

and implies that the main results of this paper hold in non-trivial analytic families of

unimodal maps depending on any number of parameters.

Chapter 5

Poincare Exponents for Fibonacci

Maps

5.1 Preliminaries

The notions of dimension that will be important for us are Hausdorff dimension, hyper-

bolic dimension and the critical exponent of the Poincare series.

Given a set X ⊂ Rn, we define the δ-Hausdorff measure by

Hδ(X) = limη→0

inf∑i

(diam Ui)δ,

where the infimum is taken over countable coverings by open sets ∪Ui of X such that

diam Ui < η for all i. We then define the Hausdorff dimension of a set X by

HD(X) = infδ : Hδ(X) = 0 = supδ : Hδ(X) is infinite.

To define the remaining two notions, we must restrict ourselves to dynamically defined

sets. Suppose that X is a compact set that is invariant under a holomorphic map that is

defined in a neighbourhood of X. Then X is called hyperbolic if there exists a constant

83

Chapter 5. Poincare Exponents for Fibonacci Maps 84

C > 0 and λ > 1 such that for all x ∈ X

|Dfn(x)| ≥ Cλn, n = 0, 1, 2, . . . .

We define the hyperbolic dimension of a set X by

HDhyp(X) = supHD(Y ) : Y is a hyperbolic subset of X.

Let O denote the post-critical set of a map f . We define the Poincare series with

exponent δ at a point z ∈ C \ O by

Θδ(f, z) =∞∑n=0

∑ζ∈f−n(z)

1

|Dfn(ζ)|δ.

The critical exponent δcr(f) is the value of δ that separates those exponents for which

the Poincare series diverges from those for which it converges:

δcr(f) = supδ : Θδ(f, z) diverges = infδ : Θδ(f, z) converges.

We define the truncated Poincare series by

Θ[j]δ (f, z) =

j∑n=0

∑ζ∈f−n(z)

1

|Dfn(ζ)|δ.

By the Koebe distortion theorem, the Poincare series has bounded oscillation on any

compact set K ⊂ C \ O:

Θδ(f, z) ≤ CΘδ(f, y), z, y ∈ K,

where the constant C depends only on the hyperbolic diameter of K in C \ O. As a

result, we see that δcr(f) is independent of the choice of z. We also have the following


estimate on the critical exponent given by straight forward length and area arguments.

Lemma 5.1.1. If J(f) is connected, then 1 ≤ δcr(f) ≤ 2.

On hyperbolic sets, the Poincare series depends continuously on f and δ:

Lemma 5.1.2. [AL3] Suppse f : X → C is a continuous map defined on a compact set X

with no isolated points that admits a holomorphic extension to a neighbourhood of X such

that f has no critical points. Then if the maximal invariant subset Q = ∩n≥0f−n(X) is

hyperbolic, then there exists δ < 2, K > 0 such that for every x ∈ X we have θδ(f, x) < K.

Moreover, δ and K can be chosen uniformly over a compact family of maps.

Let us introduce some notation to ease the exposition. When working with the

Poincare series, it will be convenient to let A ←− B denote the collection of all finite

backward orbits that start in B and end in A. We will enhance this notation by let-

ting A ←−CB denote orbits that start in B, travel through C and finish in A; that is,

(x0, x1, . . . , xk) ∈ A←−CB if x0 ∈ B, xi ∈ C for 1 ≤ i ≤ k−1 and xk ∈ A. We let A

+←− B

denote non-trivial orbits (orbits with length at least 1). Finally, when extra clarity is

needed, we will indicate that orbits under the map f are being considered by writing

Af←− B.

Combining this notation with our notation for the Poincare series, we will write:

Ξδ(Ag←−CB) = sup

z∈A

∞∑k=0

∑(x0, x1, . . . xk = z) ∈

Ag←−CB

1

|Dgk(x0)|δ

and

Ξ[j]δ (A

g←−CB) = sup

z∈A

j∑k=0

∑(x0, x1, . . . xk = z) ∈

Ag←−CB

1

|Dgk(x0)|δ.


Lemma 5.1.3. The ”chain rule” for the Poincare series:

Ξδ(A← B ← C) ≤ Ξδ(A← B)Ξδ(B ← C)

Proof. First of all, notice that every orbit (x0, . . . , xk) ∈ A← B ← C can be decomposed

into two orbits (x0, . . . , xj) ∈ A ← B and (xj, . . . , xk) ∈ B ← C, with 0 < j < k, and

that we can express

1

Dfk(x0)=

1

Df j(x0)Dfk−j(xj).

So that

Ξδ(A← B)Ξδ(B ← C)

= sup∞∑k=0

∑(x0, . . . , xk)

∈ A← B

1

|Dfk(x0)|δsup

∞∑i=0

∑(y0, . . . , yi)

∈ B ← C

1

|Df i(y0)|δ

≥∞∑k=0

∑(x0, . . . , xk)

∈ A←−BC

1

|Dfk(x0)|δ

If f is a non-renormalizable map with recurrent critical point, and all fixed points

repelling, so that we can construct the Yoccoz puzzle at all levels as usual we let

V 0 ⊃ V 1 ⊃ V 2 ⊃ . . .

be the principal nest. We let An = V n \ V n+1 denote the principal moduli. Let Dn

denote the domain of the of the return map to V n. We will enumerate the components

of the domain of the return map to V n by V n0 , V

n1 , V

n2 , . . . . We will set V n

0 = V n+1 and

we will let V n1 denote the component of the domain of the return map that contains the

critical value. We let Dn∗ = Dn \V n+1 be the collection of non-central components of the


domain of the return map. Let Qn denote the closure of the set of points that remain

in Dn∗ for all time. It is easy to see that HDhyp(f) = supn HD(Qn). We will let En be

the precritical components of the domain of the return map to V n, so that if W is a

component of En, then gn|W = gn−1|W : W → V n.

Let us also introduce the scaling factors

λn = supW∈Dn

diam W

diam V n.

We will collect some useful facts about non-renormalizable maps below.

Lemma 5.1.4 ([L2],[S1]). The Julia set of of a non-renormalizable quadratic map has

measure zero.

Using coverings of Qn given by univalent pullbacks of the components of Dn∗ , we can

prove

Lemma 5.1.5.

HD(J(g)) ≤ supn

infδδ : Ξδ(D

n∗

gn←−−Dn∗

Dn∗ ) converges.

Proof. By the previous lemmas,

HD(J(f)) = HDhyp(J(f)) = supn

HDhyp(Qn),

so, we need only show that

supn

HDhyp(Qn) ≤ supn

infδδ : Θδ(D

n∗

gn←−−Dn∗

Dn∗ ) converges.

First of all, we will estimate of the Poincare series in terms of diameters puzzle pieces.

Suppose that (x0, x1, . . . xk) ∈ Dn∗ ←−−

Dn∗Dn∗ . Let W1 denote the component of the domain


of the return map containing xk−1, W2 denote the component of Dn∗ containing xk−2, and

in general Wk−i denotes the component containing xi, 0 ≤ i ≤ k. Then

|Dgkn(x0)| = |Dgn(xk−1)||Dgn(k − 2)| . . . |Dgn(x0)|

diam V n

diamW1

diam V n

diam W2

. . .diam V n

diamWk

Let us index the components of the pullbacks of V n through iterates of gkn by W ki .

For any i we may write

diam W ki diam V ndiam W1

diam V n

diam W2

diam V n. . .

diam Wk

diam V n diam V n

|Dgkn(x0)|.

Then

Ξδ(Dn∗

gn←−−Dn∗

Dn∗ ) =

∞∑k=0

∑(x0,...,xk)

1

|Dgkn(x0)|δ

∞∑k=0

∑i

(diam W k

i

diam V n

)δ

=1

(diam V n)δ

∞∑k=0

∑i

(diam W ki )δ

Now, suppose that δ > supn infδδ : Ξδ(Dn∗

gn←−−Dn∗

Dn∗ ) converges. Then Ξδ(D

n∗

gn←−−Dn∗

Dn∗ ) converges for all n. We will show that supn HDhyp(Qn) ≤ δ.

Suppose that this is not the case. Then for any 0 < ε < δ there exists a level n such

that supn HDhyp(Qn) > δ − ε. In other words:

supν>0

inf∑

(diam(Wi)δ−ε : ∪Wi ⊃ Qn, diam Wi < ν

is infinite.

But, we can construct coverings of Qn by ∪Wi with diam Wi arbitrarily small by

taking the Wi to be long pullbacks of components of V n through orbits Dn∗ ← Dn

∗ :

choose any ν > 0, and choose k so that for any component W of Dn∗ , if we pull W back

through any univalent branch of the return map to V n k times, then diam f−kn (W ) < ν.


Then

supk∑i

(diam W ki )δ−ε

is infinite.

But since δ > supn infδδ : Ξδ(Dn∗

gn←−−Dn∗

Dn∗ ) converges, we have that Ξδ−ε(D

n∗

gn←−−Dn∗

Dn∗ )

must converge when ε is small enough, so we have a contradiction.

Let

sn(δ) = Ξδ(Dn∗

gn,+←−−Dn∗

Dn∗ ) = sup

z∈Dn∗

∞∑k=1

∑(x0, . . . , xk = z) ∈

Dn∗gn,+←−−−Dn∗

Dn∗

1

|Dgkn(x0)|δ,

and

ln(δ) = Ξδ(Vn+1 gn←−−

Dn∗Dn∗ ) = sup

z∈Dn∗

∞∑k=1

∑(x0, . . . , xk = z) ∈

V n+1 gn,+←−−−Dn∗

Dn∗

1

|Dgkn(x0)|δ.

Their truncated versions have the obvious definitions.

5.2 Quadratic Fibonacci maps

From now on g will be a Fibonacci map. As usual V 0 ⊃ V 1 ⊃ V 2 ⊃ . . . will denote its

principal nest and An = V n \ V n+1 will be the principal annuli. At each level n there

are two components of the domain of the return map that intersect the post-critical set;

let V n0 = V n+1 denote the one containing the critical point and let V n

1 denote the other

component. Let Dn denote the full domain of the return map to V n and let Dn∗ denote

the collection of non-central components of Dn. Let Qn denote the closure of the set of

points whose orbits stay in Dn∗ for all time. Let En+1 = V n+1

1 ∪ −V n+11 ⊂ Dn+1 denote

the precritical components under gn+1. A component W of En+1 is mapped on to V n+1

by gn+1|W = gn|W .

Let Un+1 = (gn|V n+1)−1(V n1 ) be the pullback of V n

1 by gn to V n+1. Let Cn+1 =


Un+1 \ V n+2 be the collar around V n+2. Notice that under gn points in the collar first

land in the non-central component, but under g2n they do not return to V n+1. Note that

gn : (V n+1, Un+1)→ (V n, V n1 ) as a double covering so that

mod(V n+1 \ Cn+1) = (1/2)mod(V n \ V n1 ).

Moreover, gn|V n+1 may be decomposed into φ sq where φ is a univalent map with

small distortion, depending on the principal moduli, and sq is the squaring map, so that

diam Un+1

diam V n+1

(diamV n

1

diam V n

)1/2

. (5.2.1)

For Fibonacci maps, we have that diam V n1 diam V n+1, which implies

(diamV n

1

diam V n

)1/2

(diamV n+1

diam V n

)1/2

. (5.2.2)

For the Fibonacci map, the scaling factors decay at an exponential rate: there exists

a constant C > 0 and λ < 1 such that λn < Cλn, and since the Fibonacci map is not

renormalizable by Lemma 5.1.4 we have that the measure of the Julia set is zero, and by

Lemma 5.1.5 δcr ≥ HD(J(f)).

5.3 Initial estimates

Since P(Dn|V n) decays exponentially [AL3], there exists n > 0 such that

sn(2) = sup Ξ2(Dn∗

gn,+←−−Dn∗

Dn∗ ) = sup

∞∑k=1

∑(x0,...,xk)

1

|Dgkn(x0)|2< ε,

where the second summation is taken over orbits (x0, . . . xk) ∈ Dn∗

gn,+←−−Dn∗

Dn∗ .


V n

V n+2

Cn+1

gn

gn V n1

V n+11

V n+1

V n+1

Figure 5.1: Fibonacci map


Since the orbits under consideration are hyperbolic by Lemma 5.1.2, we can perturb

2 to δ so that sn(δ) < ε with δ < 2.

We now estimate the value of the Poincare series for the the landing map ln from V n

to V n+1 using the chain rule for the Poincare series and the fact that when n is large the

distortion on each component of the return map to V n is small.

ln(δ) = Ξδ(Vn+1 gn←−−

Dn∗Dn∗ ) = sup

∞∑k=1

∑(x0, x1, . . . , xk) ∈

Dn∗ ]gnDn∗

1

|Dgkn(x0)|δ

= supxk

∞∑k=1

∑(x0, x1, . . . xk) ∈

Dn∗ ]gnDn∗

1

|Dgn(gk−1n (x0))|δ|Dgk−1

n (x0)|δ

≤ supxk

∑(xk−1,xk)

1

|Dgn(xk−1)|δsupxk−1

∞∑k=1

∑(x0, . . . , xk−1) ∈

Dn∗ ]gnDn∗

1

|Dgk−1n (x0)|δ

≤ ε(1 + ε),

where the 1 accounts for the trivial orbits.

5.4 Estimates for the Poincare series for the return

map

In this section we will estimate the values of the Poincare series for the return map.

Lemma 5.4.1.

Ξδ(Dn+1∗

gn,+←−−Dn∗

Dn+1∗ ) ≤ Cln(δ)λδ/2n + Cln(δ)λδ/2n + Cλδn+1

We can decompose the orbits in a single iteration of the return map according to


where they start.

Ξδ(Dn+1∗

gn,+←−−Dn∗

Dn+1∗ ) = Ξδ(D

n+1∗

gn,+←−−Dn∗

(Dn+1∗ \ (Cn+1 ∪ En+1))

+Ξδ(Dn+1∗

gn,+←−−Dn∗

Cn+1) + Ξδ(Dn+1∗

gn,+←−−Dn∗

En+1)

5.4.1 Starting inside of the collar

Let us bound Ξδ(Dn+1∗

gn,+←−−Dn∗

Cn+1). First of all, notice that all orbits starting in Cn+1

first pass through V n1 under gn and then land in Dn

∗ under a second iterate of gn. The

fact that they do not return to V n+1 under g2n is guaranteed by the fact that they are in

the collar.

So we have that

Ξδ(Dn+1∗

gn,+←−−Dn∗

Cn+1) =∞∑k=3

∑(x0,...,xk)

1

|Dgkn(x0)|δ

≤∞∑k=3

∑(x2,...,xk)

1

|Dgk−2n (x2)|δ

supx0∈Cn+1

2

|Dg2n(x0)|δ

= ln(δ) sup2

|Dg2n(x0)|δ

. (5.4.1)

We now estimate the derivative of g2n : Cn+1 → V n. First of all, recall that we can

decompose gn|V n+1 = φ sq where sq is the squaring map and φ is a univalent map

with small distortion. Thus, we can write g2n = (gn|V n

1 ) φ sq. Let x1 = g(x0) and let

c = gn(0). For the next two estimates, those for the derivatives of the maps with small

distortion, we approximate the derivative by the slope of the secant line through two

conveniently chosen points. Since gn|V n1 has small distortion,

|D(gn|V n1 )(x0)| |gn(x1)− gn(c)|

|x1 − c|=|g2n(x0)− g2

n(0)||gn(x0)− gn(0)|

,


and since φ has small distortion, we have

|Dφ(sq(x0))| |gn(x0)− gn(0)||x0|2

.

Hence

|Dg2n(x0)| = |D(gn|V n

1 (x1))Dφ(sq(x0))Dsq(x0)|

|g2n(x0)− g2

n(0)||gn(x0)− gn(0)|

|gn(x0)− gn(0)||x0|2

2|x0|

|g2n(x0)− g2

n(0)||x0|

≥ dist(∂V n+1, ∂V n+11 )

diam Cn+1 diam V n+1

diam Cn+1.

Applying (5.2.1) and (5.2.2) we have that

1

|Dg2n(x0)|δ

≤ Cλδ/2n . (5.4.2)

Therefore, combining (5.4.1) and (5.4.2) we have

Ξδ(Dn+1 gn,+←−−

Dn∗Cn+1) ≤ Cln(δ)λδ/2n . (5.4.3)

The constant C depends only the modulus and the degree of the map and remains

bounded as the moduli grows.

5.4.2 Starting outside of the collar

We will estimate

Ξδ(Dn+1∗

gn,+←−−Dn∗

Dn+1∗ \ (Cn+1 ∪ En+1))

.

In this case we need only separate a single iterate of gn from the return map:


Ξδ(Dn+1∗

gn,+←−−Dn∗

Dn+1∗ \ (Cn+1 ∪ En+1)) =

∞∑k=2

∑(x0,...,xk)

1

|Dgkn(x0)|δ

≤∞∑k=2

∑(x1,...,xk)

1

|Dgkn(x1)|δsup

x0∈Dn+1∗ \(Cn+1∪En+1)

2

|Dgn(x0)|δ

≤ ln(δ) supx0∈Dn+1

∗ \(Cn+1∪En+1)

2

|Dgn(x0)|δ(5.4.4)

We can arrive at an estimate for the derivative of |Dgn| as follows. gn = φ sq can

be decomposed into the composition of a quadratic map sq and a univalent map φ with

small distortion for large n.

So

|Dgn(x0)| = |Dφ(x0)||Dsq(x0)| ≥ Cdiam V n

(diam V n+1)2diam Cn+1 diam V n

diam V n+1

diam Cn+1

diam V n+1.

By 5.2.1 and 5.2.2,

|Dgn(x0)| ≥ Cdiam V n

diam V n+1

(diamV n+1

diam V n

)1/2

= C

(diam V n

diam V n+1

)1/2

. (5.4.5)

Consequently by (5.4.4) and (5.4.5),

Ξδ(Dn+1∗

gn,+←−−Dn∗

Dn+1∗ \ (Cn+1 ∪ En+1)) ≤ Cln(δ)λδ/2n . (5.4.6)

The constant C remains bounded as the moduli tend to infinity.


5.4.3 Starting in precritical components

Since any component W of En+1 is mapped univalently over V n+1 by gn, this map extends

to a univalent map onto Cn. Moreover,

mod(Cn \ V n+1) =1

2mod(V n−1

1 \ gn−1(V n+1)) =1

2mod(V n−1 \ V n),

since gn−1 maps (Cn+1, V n) as a double covering onto (V n−11 , gn−1(V n+1), and the return

map to V n−1 maps (V n−11 , gn−1(V n+1) univalenly onto (V n−1, V n). So the distortion of

gn|W is very small.

Notice that Ξδ(Dn+1∗

gn,+←−−Dn∗

En+1) ≤ 2 supx∈En+11

|Dgn(x)|δ . Then, the discussion in the

previous paragraph implies that

2 supx0∈En+1

1

|Dgn(x0)|δ≤ C

( diam W

diam V n+1

)δ≤ Cλδn+1.

Notice that C depends only on the degree and on the moduli, and improves as the moduli

grows.

As a result, we have that

Ξδ(Dn+1∗

gn,+←−−Dn∗

En+1) ≤ Cλδn+1. (5.4.7)

Summing up these three estimates yields the lemma stated at the start of the section.

5.5 Inductive Estimate for the Poincare Series

We first split off one iterate of the first return map:

sn+1(δ) = Ξδ(Dn+1∗

gn+1,+←−−−−Dn+1∗

Dn+1∗ ) = Ξδ(D

n+1∗

gn+1←−−−Dn+1∗

Dn+1∗

gn,+←−−Dn∗

Dn+1∗ ) (5.5.1)


≤ (Ξδ(Dn+1∗

gn+1←−−−Dn+1∗

Dn+1∗ ) + 1)Ξδ(D

n+1∗

gn,+←−−Dn∗

Dn+1∗ )

= (sn+1(δ) + 1)Ξδ(Dn+1∗

gn,+←−−Dn∗

Dn+1∗ ).

By Lemma 5.4.1

Ξδ(Dn+1∗

gn,+←−−Dn∗

Dn+1∗ ) ≤ Cln(δ)λδ/2n + Cln(δ)λδ/2n + Cλδn+1,

≤ Cln(δ)λδ/2n + Cλδ/2n ,

for n large enough, since for the Fibonacci map, the scaling factors decay exponentially.

Let us do induction on the sn(δ). For any ε we choose, we can find a level n of the

puzzle so that sn(δ) < ε for some δ < 2, and Cλδ/2n+i <

ε1−ε3 for all i ≥ 0. This establishes

the base for our induction. Now assume that we have sn+k−1(δ) < ε. Then

s[j]n+k(δ) ≤ (s

[j]n+k(δ) + 1)Ξδ(D

n+k∗

gn+k−1,+←−−−−−Dn+k−1∗

Dn+k∗ )

= (s[j]n+k(δ) + 1)(Cln+k−1(δ)λ

δ/2n+k−1 + Cλ

δ/2n+k−1)

≤ (s[j]n+k(δ) + 1)Cλ

δ/2n+k−1(ε+ ε2 + 1)

≤ (s[j]n+k(δ) + 1)

ε

1− ε3(ε+ ε2 + 1)

= (s[j]n+k(δ) + 1)

ε+ ε2 + ε3

1− ε3

So we get that

s[j]n+k(δ) ≤

ε+ε2+ε3

1−ε3

1− ε+ε2+ε3

1−ε3=

ε+ ε2 + ε3

1− ε3 + ε+ ε2 + ε3= ε.

This completes the induction. Letting j go to infinity we see that all the sn(δ) are

less than ε and by Lemma 5.1.5 HD(J(f)) < 2.

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