Randomness in Hyperbolic Dynamics

Randomness in hyperbolic dynamics

Jamerson Bezerra

Rio de Janeiro July 20, 2020 Instituto Nacional de Matematica´ Pura e Aplicada

Jamerson Bezerra

RANDOMNESS IN HYPERBOLIC DYNAMICS

Thesis presented to the Post-graduate Program in Mathe- matics at Instituto Nacional de Matematica´ Pura e Aplicada as partial fulﬁllment of the requirements for the degree of Doctor in Philosophy in Mathematics.

Advisor: Marcelo Viana Co-advisor: Carlos Gustavo Moreira

Rio de Janeiro 2020 “The ﬁrst rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.” George Polya´ ACKNOWLEDGMENTS

I would like to express my gratitude to my advisor, Professor Marcelo Viana, for the enlightening conversations and for his support throughout the entire process of construction of this work. I would also to thank my co-advisor, Professor Carlos Gustavo Moreira (Gugu), for his persistence in convince me to work with the second part of this thesis and for accept me as his student in the last months of my Ph.D. period. Without him this work would not be possible. I would like to reinforce that Viana and Gugu’s support and presence during the pandemic was essential to the conclusion of this work and for that I am really thankful. I am particular grateful to, my collaborator and friend, Professor Mauricio Poletti, for accept to work with me, receive me at Orsay where we were able to finish the first part of this thesis, and for be one of the responsible in my choice of dynamical systems as my work area. I would like to offer my special thanks to Sandoel Vieira, Thomas´ Spier, Hugo Araujo,´ Sergio Ibarra and Ivan Passoni for them availability and patience to hear me and for all the important commentaries and observations about the work without which this thesis would not be the same. I wish to acknowledge the help of Gregory Cosac, Fernando Lenarduzzi and Marlon Flores. In uncountable moments I disturb them, among other things, with corrections of English, review of my projects and works and without think twice they were always available to help me with these tasks. For that I have no words to say but thank you. I would like to specially thanks my friends that help me in my academic journey for the countless advices and inspirations. Just to mention some of them: Alan Anderson, Andres´ Chavez,´ Argenis Garcia, Caio Doria,´ Davi Lima, Eduardo San- tana, Felipe Gonçalves, Luis Fernando, Mateus Sousa, Paulo Ribenboim, Pedro Gaspar,´ among others. I admire you and you are example for me. I would like to thanks my awesome wife, M´ıriam Pessoa.ˆ She gave me all the support that a needed to walk through this path. Her patience to deal with me can not be found in another human been. She gave me countless meaningful advices in numerous situations and for this reason she is as important as me for this thesis. There is not a way to compile a set words in such a way to express my gratitude to her. I only can say thank you my dear. I would like to say thank you to my mother and my father to raise me, in very adverse circumstances, prioritizing education at every step of the path. They always believed that the unique way to have better opportunities in the life is through education and so gave me all the necessary conditions that bring me until here. Finally, I wish to thanks all the friends that I made during this journey. Beyond all the above mentioned I would like to mention: Marcos, Gisele and Roberto, Clea and Maxwell, Thais and Alcides, Eduardo Garcez, Eduardo dos Santos, Vi- tor, Ermerson and many others. Thank you all.

4 ABSTRACT

This thesis is a contribution to the widely studied theory of hyperbolic dynamics. The work is structured in two independent parts which are intrinsically related: Lyapunov exponents for linear cocycles (model for non-uniformly hyperbolic dynamics) and geometric properties of horseshoes (uniformly hyperbolic dynamics). In the ﬁrst part, we address the problem of continuity and simplicity of the Lyapunov spectrum for random product k-tuples of quasi periodic cocycles. In dimension two, we prove that for any r ≥ 1 there exists a C0 open and Cr dense set of k-tuples of Cr quasi periodic cocycles whose random product is a continuity point of the Lyapunov exponents with positive value. Restricting to the k-tuples of Schrodinger¨ cocycles, the same results holds. In higher dimensions, we prove that among the Cr, r ≥ 1, k-tuples of quasi periodic cocycles with one coordinate being a diagonal cocycle, there exists a Cr dense and C1 open set such that the random product deﬁned by cocycles in this set has simple Lyapunov spectrum and is a C0 continuity of the Lyapunov exponents. In the second part of this work, using the Erdos¨ probabilistic method, we prove that for typical Cr horseshoes with s- splitting, one dimensional weak stable bundle and upper stable dimension smaller than one there exists a subhorseshoe with almost the same upper stable dimension which is contained in a C1+ locally invariant submanifold tangent to the center unstable direction.

Key-words: Lyapunov exponents. Quasi-periodic cocycles. Higher dimensional horseshoes. Reduction of the dimension. CONTENTS

Acknowledgments 3

Abstract 5

I Random product of quasi-periodic cocycles 9

1 Introduction 10

2 Deﬁnitions and Statements 12

3 Holonomies 15

4 Proof of the results 18 4.1 Proof of Theorem A ...... 18 4.2 Proof of Theorem B ...... 21 4.3 Proof of Theorem C ...... 23 4.4 Proof of Proposition I.10 ...... 26

II Embedding small horseshoes in lower dimensions 32

5 Introduction 33 5.1 History and motivation ...... 33 5.2 Results ...... 36 5.3 Work in progress ...... 37

6 Horseshoes 39 6.1 Dominated splitting and partially hyperbolic sets ...... 39 6.2 Horseshoes ...... 41 6.3 Invariant manifolds ...... 44 6.4 Markov partitions ...... 46

6 6.5 Symbolic dynamics ...... 47 6.6 Symbolic dynamics notation ...... 48

7 Metric interactions between horseshoes and its symbolic models 53 7.1 Stable diameters ...... 54 7.2 Upper dimensions ...... 56

8 Horseshoes with s-splitting and one dimension weak stable bundle 60 8.1 Bounded distortion estimates ...... 61 8.2 Applications of bounded distortion ...... 64 8.3 Counting pieces ...... 68 8.4 Extractions ...... 72

9 Random perturbations and projections 76 9.1 Random perturbations ...... 77 9.2 Translating randomly in the weak stable directions ...... 78 9.3 Nonrecurrence and transversality condition ...... 81 9.4 Marstrand like argument ...... 85

10 Proof of the main theorem 89

7 LIST OF FIGURES

3.1 Stable holonomy ...... 16 3.2 Linear stable holonomy ...... 16

3.3 Holonomy Ht ...... 16

4.1 Weak twisting condition...... 19 4.2 Example of algebraic set. Cartan umbrella given by the equation z(x2 − y2) − x3 = 0...... 26 1 1 4.3 Transversality between V0 and A(S ) = {Ht; t ∈ S }...... 28

0 5.1 Λg embedded in S ...... 37

6.1 Smale horseshoe...... 41 6.2 Homoclinic intersection in dimension 2...... 42 6.3 s-splitting...... 42 6.4 Non s-splitting...... 43 6.5 Markov Partition with small diameter...... 47 6.6 From stable cylinder to unstable cylinder...... 52

7.1 Piece P(a) with diameter larger than ζ...... 54 7.2 Stable diameter ...... 55

9.1 Random perturbations ...... 79 9.2 Projection of the boxes ...... 85

10.1 Picking boxes with disjoint projections...... 89

8 Part I

Random product of quasi-periodic cocycles

9 1 INTRODUCTION

When studying the Lyapunov exponents of linear cocycles two main questions appears frequently: Are the exponents continuous with respect to the cocycle and how frequently do we have the maximum number of different exponents? We say that the Lyapunov spectrum of a cocycle is simple if it has the maximum number of different exponents (this is the case when all Oseledets spaces are one dimensional). If we deal with two dimensional cocycles, this means our matrices take values in SL2(R), much more is known than in the higher dimensional case, when it takes values in GLd(R), d > 2. It was proved by Bochi [8] that in the C0 topology generically two dimensional cocycles either have uniform hyperbolicity or have only one exponent. An- other result, by Avila [4], says that in the smooth topology there is a dense set of Cr-cocycles for 0 ≤ r ≤ ∞ with simple spectrum. In particular these two results imply that in the C0 topology the exponents do not behave continuously, moreover, the only continuity points are the hyperbolic and cocycles with only one exponent. This motivates the following question: Are there open and dense sets of cocycles with simple spectrum, or, in other words, is the simplicity of the spectrum a generic property in smooth topology? For two dimensional cocycles over bases with some hyperbolic behaviour this was proved in many scenarios (see [48], [1], [42]). For two dimensional smooth quasi-periodic cocycles, (Cr topology, 0 < r ≤ ∞) Wang and You proved in [50] that the set with simple spectrum is not open, in particular the exponents are not continuous with respect to the cocycle. For higher dimensional cocycles the problem of simplicity becomes more delicate, this problem goes back to the works of Guivarc’h-Raugi [24] and Gol’dsheid- Margulis [22], where they work with random product of matrices. By random product of matrices we mean the cocycle generated by composing randomly a set of matrices accordingly to a probability in the group of matrices. They prove simplicity of the Lyapunov spectrum in this scenario with some generic conditions on the support of the probability measure.

10 For more general higher dimensional cocycles over hyperbolic maps, if some bunching conditions are assumed, simplicity was proved to be generic in the Cr topology, for 0 < r < ∞ ([10], [2], [6]). For quasi-periodic cocycles very few is known in higher dimension about simplicity, see for example [18] where they find some strong conditions to have simplicity of the biggest Lyapunov exponents. In this work we study the Lyapunov exponents of cocycles over dynamics that have both behaviors, a random part and a isometric (quasi periodic) part. Specifically we deal with random product of quasi-periodic cocycles by this we mean the cocycle generated by taking a probability measure in the set of quasi- periodic cocycles and iterating randomly according to this probability. We prove that, for two dimensional cocycles, there exists an open and dense set of cocycles that are points of continuity for the Lyapunov exponents and have simple spectrum, even in the C0 topology, and in higher dimension, with one of them taking values in the diagonal group, we find open and dense sets with simple spectrum. Observe that in contrast to the non-random case, where, in the C0 topology, generically we have one Lyapunov exponent outside of uniform hyperbolicity, in the random product scenario we get generically positive Lyapunov exponents also in the C0 topology. This change of behavior when we ad some randomness on the dynamics was already exploited in many cases, see for example [47], [27], [7].

11 2 DEFINITIONS AND STATEMENTS

Given an invertible measurable map f : M → M and a measurable application d d A : M → GLd(R), we deﬁne the linear cocycle as the map F : M × R → M × R given by

F(x,v) = ( f (x),A(x)v).

Usually we denote the linear cocycle by the pair ( f ,A) and, sometimes, when the map f is ﬁxed (and there is no ambiguity) we denote just by A. Its iterates are given by Fn(x,v) = ( f n(x),An(x)v), where  A( f n−1(x))···A(x), if n ≥ 1  An(x) = Id, if n = 0  A( f n(x))−1 ···A( f −1(x))−1 if n ≤ 1 In the case that f preserves a probability measure µ which is ergodic and ± log||A 1|| is µ-integrable, Oseledets theorem [37] says that there exists k ∈ N, real d 1 k numbers λ1(A) > ... > λk(A) and a decomposition of R = E (x)⊕···⊕E (x) by measurable subspaces Ei(x) such that for µ-a.e. x ∈ M we have

• A(x)Ei(x) = Ei( f (x)) for all i, and

n i • lim ||A (x)v|| = λi(A), for all v ∈ E (x). |n|→∞

The numbers λ1(A),··· ,λk(A) are called the Lyapunov exponents associated with the cocycle ( f ,A) and the set formed by them is called Lyapunov spectrum, we say that A has simple spectrum if k = d. When M = S1 and f is a rotation of angle θ ∈ (0,1] we say that the cocycle ( f ,A) is a quasi-periodic cocycle and usually write (θ,A). k Fix θi ∈ (0,1] for all i ∈ Ik = {0,...,k} and ν = ∑i=0 νiδi a probability measure 1 Z ˆ on Ik. We will denote by Leb, the Lebesgue measure on S . Let X = Ik , X = X × S1 and consider the invertible (locally constant) skew product fˆ : Xˆ → Xˆ , given by ˆ f ((xn)n∈Z,t) = ((xn+1)n∈Z,t + θx0 )

12 1 and observe that the measure µˆ = νZ ×Leb is fˆ-invariant. We also write fi : S → 1 f (t) = t + f n = f ◦···◦ f f −n = f −1 ◦···◦ f −1 n ∈ S , i θi, x xn−1 x0 and x x−n x1 , for all N and x ∈ X.

From now on we will fix θ0 irrational. In particular, we have that ( fˆ, µˆ ) is an ergodic system. 1 For measurable maps Ai : S → GLd(R), i ∈ Ik, we define the random product ˆ ˆ ˆ ˆ of the quasi periodic cocycles (θi,Ai)i∈Ik as the cocycle ( f ,A), where A : X → ˆ GLd(R) is given by A(x,t) = Ax0 (t). r 1 k+1 Note that each vector (A0,...,Ak) ∈ (C (S ,GLd(R))) defines a random product ( fˆ,Aˆ). We abuse of the notation and denote by Aˆ the above map and the

point (A0,...,Ak) which deﬁne Aˆ. Let 0 ≤ r ≤ ∞ (or r = ω for the analytic case), by Cs topology in the product r 1 k+1 space (C (S ,GLd(R))) , for s ∈ [0,r] (or s ∈ [0,∞] ∪ {ω} in the case when r = ω) we mean the topology given by the distance

d s (Aˆ,Bˆ) = maxd s 1 (A ,B ), C C (S ,GLd(R)) i i i∈Ik

r 1 k+1 where Aˆ,Bˆ ∈ (C ( ,GL ( ))) and d s 1 denote the distance that gen- S d R C (S ,GLd(R)) s r 1 erates the C topology in C (S ,GLd(R)). r 1 k+1 s We say that the Aˆ ∈ (C (S ,GLd(R))) is a C -continuity point for the Lya- r 1 punov exponents if for all sequences Aˆk ∈ C (S ,GLd(R)) converging to Aˆ in the s C -topology we have that the Lyapunov exponents of Aˆk converge to the Lyapunov exponents of Aˆ. These cocycles where already introduced in [5, section 5.2], to show that cocycles over some partially hyperbolic dynamics are not continuous in general. If our cocycle A takes values in SL2(R) recall that we have λ1(A) = −λ2(A), in particular we have simple spectrum if and only if we have one positive exponent. If the cocycle takes values in the space of two by two matrices with positive determinant, we can reduce to SL2(R) just dividing by the square root of the determinant.

Theorem A. For r ∈ [0,∞] ∩ {ω}, there exists a C0 open and Cr dense subset r 1 k+1 of (C (S ,SL2(R))) , such that the random product deﬁned by cocycles in this set has positive Lyapunov exponent and is a C0-continuity point for the Lyapunov exponents.

Let ϕ : S1 → R be a continuous function. The Schrodinger cocycle associated 1 1 1 to the function ϕ is deﬁned as ( f ,Aϕ ), where f : S → S and Aϕ : S → SL2(R) is given by

13 ! ϕ(x) −1 Aϕ (x) = . 1 0

In the literature it is common to use the function ϕ(x) = E −u(x), with E ∈ R and u : S1 → R a continuous function. The reason for that notation is the relation of 2 2 the Schrodinger cocycles with the Schrodinger operator Hu,x : l (Z) → l (Z), n (Hu,x(z))n = zn+1 + zn−1 + u( f (x))zn, given by the eigenvalue equation

Hu,x(z) = E · z. For a detailed survey on this topic see [17]. 1 When f is a rotation of angle θ ∈ (0,1] in S , Schrodinger cocycles, ( f ,Aϕ ), are quasi periodic cocycles which we will denote just by (θ,Aϕ ). Theorem B. For r ∈ [0,∞] ∪ {ω}, there exists a C0 open and Cr dense subset r k+ of (C (S1,R)) 1, for 0 ≤ r ≤ ∞, such that the random product deﬁned by the Schrodinger cocycles associated with the functions in this subset has positive Lya- punov exponent and is a C0-continuity point for the Lyapunov exponents.

In order to state the result in higher dimensions let Dd(R) be the subgroup of diagonal matrices in GLd(R). Theorem C. For d > 2 and r ∈ [0,∞] ∪ {ω}, there exists a Cr dense subset of r 1 r 1 k C (S ,Dd(R)) × (C (S ,GLd(R))) such that the random product deﬁned by cocycles in this set has simple Lyapunov spectrum and is a C0-continuity point of the Lyapunov exponents. If r ∈ [1,∞] ∪ {ω} this set is also C1 open. Observe that, since the Cr dense set in the Theorem C is formed by cocycles with simple Lyapunov spectrum and C0-continuity points of the Lyapunov exponents, we have a C0 open and Cr dense set with simple Lyapunov spectrum for each r ∈ [0,∞] ∪ {ω}. 1 1 1 Remark 1. We say that two cocycles A,B ∈ C(S ,GLd(R)) over f : S → S , are r r 1 C -cohomologous if there exists C ∈ C (S ,GLd(R)) such that A(t) = C−1( f (t))B(t)C(t).

Cohomologous cocycles have the same Lyapunov exponents. As a consequence,

Theorem C, is also valid for cocycles such that A0 belongs to the set of cocycles r C -cohomologous to cocycles taking values in Dd(R).

14 3 HOLONOMIES

Given x ∈ X, we deﬁne its stable set as

s W (x) = {y ∈ X, such that for some k ≥ 0, yi = xi for i ≥ k},

and the unstable set as

u W (x) = {y ∈ X, such that for some k ≤ 0, yi = xi for i ≤ k}.

We write x ∼s y if x and y are in the same stable set and similarly we write x ∼u y if x and y are in the same unstable set. s s 1 1 If x ∼ y we deﬁne the stable holonomy from x to y, hx,y : S → S , as

s n −1 n n0 −1 n0 hx,y = lim ( fy ) ◦ fx = ( fy ) ◦ fx . n→∞

where n0 is the smallest integer such that xi = yi for all i ≥ n0. Analogously, we u u 1 1 deﬁne for x ∼ y the unstable holonomy from x to y, hx,y : S → S , as

u n −1 n n0 −1 n0 hx,y = lim ( fy ) ◦ fx = ( fy ) ◦ fx . n→−∞

where n0 is the biggest integer such that xi = yi for all i ≤ n0.. Now considerx ˆ,yˆ ∈ Xˆ ,x ˆ = (x,t) andy ˆ = (y,t0). We writex ˆ ∼s yˆ if x ∼s y and 0 s u u 0 u t = hx,y(t) and we writex ˆ ∼ yˆ if x ∼ y and t = hx,y(t). Deﬁne the Linear Stable Holonomies associated with the random product ( fˆ,Aˆ) as the family of linear maps

s,Aˆ d d ˆ s {Hxˆ,yˆ : R → R ;x ˆ,yˆ ∈ X,xˆ ∼ yˆ},

given by

ˆ s,A n 0 −1 n n0 0 −1 n0 H 0 = lim (Aˆ (y,t )) Aˆ (x,t) = (Aˆ (y,t )) Aˆ (x,t). (x,t)(y,t ) n→∞ where n0 ≥ 1, is such that xi = yi for i ≥ n0. Analogously we deﬁne the Linear Unstable Holonomies

u,Aˆ d d ˆ u {Hxˆ,yˆ : R → R ;x ˆ,yˆ ∈ X,xˆ ∼ yˆ}.

15 Figure 3.1: Stable holonomy Figure 3.2: Linear stable holonomy

∗ ∗,Aˆ We use the notations ∼ or Hxˆ,yˆ , meaning that the sentence remains true for any ∗ ∈ {s,u}. Observe that for ﬁxedx ˆ,yˆ ∈ Xˆ withx ˆ ∼∗ yˆ the map

ˆ 0 1 k+1 ∗,Aˆ A ∈ (C (S ,GLd(R))) 7→ Hxˆ,yˆ varies continuously in the C0 topology. Now we will ﬁx some notations that will be essentials in the following sections.

From now on, p ∈ X will be the fix point of the shift map defined by pi = 0 for all i ∈ Z, z ∈ X will the homoclinic intersection point defined by z0 = 1 and 0 u zi = 0 for all i 6= 0. Let t = hp,z(t) and define

s u Aˆ s,Aˆ u,Aˆ h = hz,p ◦ hp,z and Ht = H(z,t0)(p,h(t)) ◦ H(p,t)(z,t0). (3.1)

Figure 3.3: Holonomy Ht

16 1 ˆ Aˆ Observe that for any ﬁxed t ∈ S the map A 7→ Ht varies continuously in the C0 topology. We omit the index Aˆ when the random product that we are dealing with is clear by the context.

17 4 PROOF OF THE RESULTS

4.1 Proof of Theorem A

In this section all cocycles takes values in SL2(R), in particular, for any map A, we have at most 2 exponents λ+(A) and λ−(A). So, in this section and in the next the term Lyapunov exponents refers to λ+ and this is not a restriction since λ+(A) = −λ−(A). ˆ Recall that the system ( f , µ) is ergodic, since we are assuming that θ0 ∈ R\Q. ˆ ˆ 0 1 k+1 Let ( f ,A) be the random product of (A0,...,Ak) ∈ (C (S ,SL2(R))) .

Deﬁnition I.1 (Weakly pinching). We say that the cocycle ( fˆ,Aˆ) is weakly pinch-

ing if the cocycle (θ0,A0) has λ+(A0) > 0 with respect to the Lebesgue measure on S1.

1 By Oseledets theorem, if λ+(A0) > 0, there exists a measurable map S 3 1 1 t 7→ (e+(t),e−(t)) ∈ RP × RP where e+(t) is the direction of the Oseledets decomposition corresponding to λ+(A0) and e−(t) the direction corresponding to λ−(A0). Take p, z, h and Ht as deﬁned in the end of Section 3.

Deﬁnition I.2 (weakly twisting). We say that a weakly pinching cocycle ( fˆ,Aˆ) is weakly twisting if

Ht({e+(t),e−(t)}) ∩ {e+(h(t)),e−(h(t))} = /0

for a positive measure subset of t ∈ S1

Observe that both conditions, weakly pinching and weakly twisting, only de-

pends on (θ0,A0) and (θ1,A1). The pinching and twisting conditions deﬁned here (and in section 4.3) are gen- eralizations of the ones introduced in [2] for cocycles over hyperbolic dynamics. We will call the cocycle ( fˆ,Aˆ) weakly simple, if it is both weakly pinching and weakly twisting.

18 Figure 4.1: Weak twisting condition.

Deﬁne the projective cocycle ˆ ˆ 1 ˆ 1 ˆ ˆ PFAˆ : X × RP → X × RP , (xˆ,[v]) 7→ ( f (xˆ),[A(xˆ)v]) ˆ and let m be an PFAˆ-invariant measure that projects on µ. Using Rokhlin’s Disin- tegration Theorem, we can ﬁnd a measurable mapx ˆ 7→ mxˆ such that Z 1 m = mxˆdµ and mxˆ({xˆ} × RP ) = 1. Xˆ We say that m is u-invariant if there is a total measure set X0 ⊂ Xˆ such that for 0 u u everyx ˆ,yˆ ∈ X withx ˆ ∼ yˆ, we have (Hxˆ,yˆ)∗mxˆ = myˆ. Analogously, we say that m is s-invariant if the same is true changing unstable by stable holonomies. Finally, m is su-invariant if it is both s an u invariant. By [36, Proposition 3.9], weakly simple implies that the projective cocycle,

PFAˆ, do not admit any su-invariant measure. We have the following criteria. ˆ r 1 k+1 Proposition I.3. Let r ∈ [0,∞]∪{ω} and take A ∈ (C (S ,SL2(R))) . If the random product ( fˆ,Aˆ) is weakly simple then, there exists a C0 neighborhood of Aˆ in r 1 k+1 (C (S ,SL2(R))) , such that any random product of cocycles in this neighborhood, has postive Lyapunov exponent and is a C0-continuity point of the Lyapunov exponent. Proof. The invariance principle of [3] (see [42, Theorem 6.2] for a version that

ﬁts into our settings) says that if λ+(Aˆ) = 0 then any m, PFˆ invariant measure, is su-invariant. Since, ( fˆ,Aˆ) is weakly simple, we conclude that λ+(Aˆ) > 0. Take Aˆ → Aˆ such that λ (Aˆ ) λ (Aˆ). Consider m , PFˆ invariant measure k + k 9 + k Aˆk that projects to µˆ , u-invariant such that

Z Aˆk(xˆ)v λ (Aˆ ) = log dm . + k kvk k

19 Up to taking a sub-sequence we can assume that

∗ mk * m and λ+(Aˆk) → a < λ+(Aˆ), ˆ where m is an PFAˆ-invariant measure that projects to µ. + As λ+(Aˆ) > 0, otherwise it is a continuity point, we have that m = αm + − ∗ R βm , with α + β = 1 where m = δe∗(xˆ)dµ(xˆ) and e∗(xˆ) is the Oseldets subspace corresponding to λ∗(Aˆ) for ∗ = + or −. So,

Z Aˆ(xˆ)v log dm = αλ (Aˆ) + βλ (Aˆ). kvk + −

By assumption we have that αλ+(Aˆ)+βλ−(Aˆ) < λ+(Aˆ), which implies that α < 1. Recall that the holonomies varies continuously in the C0 topology, then for ∗ ∗,Aˆk ∗,Aˆ everyx ˆ ∼ yˆ with ∗ = s or u, we have that Hxˆ,yˆ converges uniformly to Hxˆ,yˆ , so we can apply [42, Theorem A.1] to conclude that m is u-invariant. Observe that − − 1 + − the measure m is s-invariant and m = β (m−αm ), then m is also u-invariant. Analogously we conclude that m+ is su-invariant. Since ( fˆ,Aˆ) does not admit su-invariant measures we conclude that Aˆ is a continuity point of the Lyapunov exponent λ+. ˆ r 1 k+1 0 Assume now that there exists a sequence Ak ∈ (C (S ,SL2(R))) , of C - discontinuity points of the Lyapunov exponent, converging to Aˆ in the C0 topology. Repeting the above argument we can see that for each k ∈ N we can find a PF -invariant measure, that projects to µˆ that is su-invariant. Passing to a subse- Aˆk quence we can find m such that mk converges to it, again by [42, Theorem A.1] ˆ m is an su-invariant measure for PFAˆ. This contradicts the fact that the random product ( fˆ,Aˆ) does admit any su-invariant measure. 0 ˆ r 1 Therefore we can find a C neighborhood of A ∈ (C (S ,SL2(R))) such that the random product defined by the cocycle in this neighborhood is a C0 continuity point of the Lyapunov exponent.

To conclude the proof of theorem A, we just need to prove that the weakly simple random products are Cr dense.

r r 1 2 Proposition I.4. For r ∈ [0,∞]∪{ω}, there exists a C dense set of C (S ,SL2(R)) such that for any (A0,A1) in this set, the random product of (θi,Ai)i∈Ik is weakly r 1 k−1 simple for all (A2,...,Ak) ∈ (C (S ,SL2(R))) . r Proof. By [4] we can ﬁnd a C -dense set of A0 such that λ+(A0) > 0. In other words we have a Cr dense set weakly pinching.

20 −1 1 Now observe that in our setting Ht = A0(h(t)) A1(t). Take K ⊂ S with 1 Leb(K) > 2 such that t 7→ (e+(t),e−(t)) is continuous in K. Thus, since h preserves the Lebesgue measure we have that Leb(K ∩ h−1(K)) > 0. Take t ∈ K ∩ h−1(K) such that t is a density point for the Lesbegue measure. If

Ht{e+(t),e−(t)} ∩ {e+(h(t)),e−(h(t))} = /0,

then, by continuity, there exists a neighborhood of t in K ∩ h−1(K), containing a set with positive measure satisfying the same property which implies that the random product is twisting.

If Ht{(e+(t),e−(t)}∩{e+(h(t)),e−(h(t)} 6= /0we change A1 by A˜1 = A1 ◦Rθ , for θ small.

This implies the holonomy of (A0,A˜1) is given by H˜t = HtRθ . Then we can take θ arbitrarily small such that

H˜t ({e+(t),e−(t)}) ∩ {e+(h(t)),e−(h(t))} = ∅.

Consequently, (θi,Ai) is weakly twisting concluding the proof.

4.2 Proof of Theorem B

In the case of Schrodinger¨ cocycles the perturbation to get weakly simple cocy-

cles is more delicate because we can only perturb ϕi, i ∈ Ik. So to conclude the Theorem B, using I.3, we just need to prove the following Proposition.

r Proposition I.5. For r ∈ [0,∞] ∪ {ω}, there exists a dense subset of (C (S1,R))2 such that, for any (ϕ0,ϕ1) on this subset, the random product of the Schrodinger r 1 k−1 cocyles (θi,Aϕi )i∈Ik , is weakly simple, for any (ϕ2,...,ϕk) ∈ (C (S ,R)) . Proof. In [4], it is proved that for any r ∈ [0,∞] there is a dense subset of maps 1 r 1 ϕ : S → R in C (S ,R) such that the cocycle (θ0,Aϕ ) has positive Lyapunov exponent, which in our case is equivalent to say that for a dense subset of ϕ0 ∈ r 1 C (S ,R), the random product of (θi,Aϕi ), with i = 0,...,k, is weakly pinching. r 1 k+1 Take a point (ϕ0,...,ϕk) ∈ (C (S ,R)) such that the random product of

(θi,Aϕ )i∈Ik is weakly pinching and assume, without loss of generality, that ϕ0 does not vanish identically (this can be made since we have density of weakly

pinching). Consider p,z,h and Ht as in Section 3 and observe that ! −1 1 0 Ht = (Aϕ0 (h(t))) Aϕ1 (t) = , (4.1) ϕ0(h(t)) − ϕ1(t) 1

21 and s u h(t) = hz,p ◦ hp,z(t) = t + (θ1 − θ0) is a rotation and, in particular, preserves the Lebesgue measure. Observe that the matrix Ht preserve the vertical axis, i.e. e2 = (0,1) is the unique ﬁxed point of the action of the matrix Ht in the projective space when ϕ1(t) 6= ϕ0(h(t)). r With a small change of ϕ1 in the C topology we can assume that ϕ0(h(t)) 6= 1 ϕ1(t) for Leb-a.e. t ∈ S (for example adding a suitable constant to ϕ1). Hence, e2 is the unique direction in the projective space which is invariant by Ht. Consider the following set

1 L2 = {t ∈ S ;e2 ∈ {e+(t),e−(t)}},

where e+ and e− are the Oseledets subspaces associated to the cocycle (θ0,Aϕ0 ) (which we know that has positive Lyapunov exponent). 1 We claim that Leb(S \L2) > 0. Indeed, otherwise, we have that e2 is a Os- 1 eledets subspace of the cocycle (θ0,Aϕ0 ) for Leb-a.e. t ∈ S . Then, we have the following possibilities:

1 1. e2 = e+(t) = e+( fθ0 (t)), for some t ∈ S : In this case we have,

e2 = e+( fθ0 (t)) = Aϕ0 (t)e+(t) = (−1,0),

which is a contradiction.

1 2. e2 = e+(t) = e−( fθ0 (t)), for Leb-a.e. t ∈ S : We have,

e = e ( f 2 (t)) = A ( f (t))A (t)e (t) = (− (t),1), 2 + θ0 ϕ0 θ0 ϕ0 + ϕ0

1 for Leb-a.e. t ∈ S . This shows that ϕ0 vanishes identically, a contradiction. 1 Therefore, let δ = Leb(S \L2) ∈ (0,1) (the case δ = 1 is simpler and follows 1 analogously). By Lusin’s theorem, there exists Γ ⊂ S , such that the functions e+ 2−δ and e− are simultaneously continuous in Γ and Leb(Γ) > 2 . Observe that the condition on the measure of Γ and the fact that h preserves the Lebesgue measure shows that,

Leb(Γ ∩ h−1(Γ)) > 1 − δ.

22 Then, −1 1 Leb(Γ ∩ h (Γ) ∩ (S \L2)) > 0. −1 1 Let t ∈ Γ ∩ h (Γ) ∩ (S \L2) be a density point for the Lebesgue measure. Since e2 ∈/ {e+(t),e−(t)} we have e2 ∈/ Ht({e+(t),e−(t)}). Assume that

Ht({e+(t),e−(t)}) ∩ {e+(h(t)),e−(h(t))} 6= ∅.

Then, changing ϕ1 in a small neighborhood of t and using (4.1) we can make

Ht({e+(t),e−(t)}) ∩ {e+(h(t)),e−(h(t))} = ∅.

Since t is a density point, Ht, e+ and e− are continuous in t, we have that the above property is preserved for a positive measure neighborhood of t. r 1 So we conclude that, ﬁxed ϕ0 ∈ C (S ,R) such that the cocycle (θ0,Aϕ0 ) r 1 has positive Lyapunov exponent, there exists a dense subset of ϕ1 ∈ C (S ,R) such that the random product of (θi,ϕi) is weakly twisting, for any ϕ2,...,ϕk ∈ r C (S1,R) and so, weakly simple.

4.3 Proof of Theorem C

From now on our cocycles take values on GLd(R), for d > 2. As always, let p,z,h and Ht be as in section 3.

Deﬁnition I.6 (Pinching). We say that the random product ( fˆ,Aˆ) is Pinching if the

Lyapunov exponents λ1(p),··· ,λl(p) of the cocycle ( f0,A0) satisﬁes that l = d and for any 1 ≤ j ≤ d − 1 and Leb-a.e. t ∈ S1 the sums

λi1 (p) + ··· + λi j (p), (4.2)

for all sequences 1 ≤ i1 < ... < i j ≤ d, are distinct.

Take I and J subsets of {1,··· ,d} with the same cardinality and consider the map PI,J : GLd(R) → R deﬁned as the determinant of the matrix obtained taking the minor associated with the lines designated by the elements in I and columns designated by elements of J.

Deﬁnition I.7 (Twisting). We say that the cocycle ( fˆ,Aˆ) is Twisting if for any I and J as above Z |log|PI,J(Ht)||dt < ∞. S1

23 Remark 2. The deﬁnition of twisting given above is adapted to suit our purposes. In general, a cocycle ( fˆ,Aˆ) is said twisting if the growth rate of all minors of

H ˆn are sub exponential, i.e., fxˆ (t)

lim log PI,J(H ˆn ) = 0, n→∞ fxˆ (t) for all I,J. This, basically, says that the angle between the Oseledets subspaces goes to zero at most sub exponentially.

We say that the random product ( fˆ,Aˆ) is Simple if it is both Pinching and Twisting. Observe that even if d = 2 being simple is a stronger condition than being weakly simple. In order to prove Theorem C we need the following result which is a version of the main result of [43].

Theorem I.8. If the cocycle ( fˆ,Aˆ) is simple, then the Lyapunov spectrum is simple and it is a continuity point with respect to the C0 topology of the Lyapunov exponents.

In [43] the result is stated for Holder¨ cocycles with some more general dynamics, but as mentioned in [43, section 4.1] we only need to have well deﬁned holonomies that varies continuously with respect to the cocycle. 1 1 Observe A0 : S → Dd(R) is deﬁned by d functions a1,...,ad : S → R such that (A0(t))i,i = ai(t) and (A0(t))i, j = 0 for i 6= j. Then, by Birkhoff’s ergodic theorem, the Lyapunov spectrum of ( f0,A0) is the set Z log(ai)dLeb .

1 So, after suitable choose of bi > 0, we can deﬁnea ˜i : S → R given bya ˜i(t) = biai(t) such that the diagonal cocycle A˜0 deﬁned usinga ˜ has the property 4.2. Moreover, this is a C0 open condition. r Hence after a C small perturbation of A0 we can assume that the random product ( fˆ,Aˆ) of (A0,A1,...,Ak) is always pinching for any (A1,...,Ak) in r 1 k C (S ,GLd(R)) , and r ∈ [0,∞] ∪ {ω}. If ( fˆ,Aˆ) is twisting, then, by Theorem I.8 we have that ( fˆ,Aˆ) has simple Lya- punov spectrum and is a C0-continuity point of all Lyapunov exponents. In partic- r 1 k+1 0 ular, any (B0,...,Bk) ∈ (C (S ,GLd(R))) which is C -close to (A0,...,Ak) has also simple Lyapunov spectrum. So, to conclude the proof of the Theorem C it is enough to prove the following theorem:

24 Theorem I.9. Let d > 2, r ∈ [0,∞] ∪ {ω}. Then the set of maps A1 in r 1 ˆ ˆ C (S ,GLd(R)), such that the random product ( f ,A) of (θi,Ai)i∈Ik is twisting, is Cr dense. Moreover for r ∈ [1,∞] ∪ {ω} this set is also C1 open.

r 1 Proof. Assume ﬁrst that r ∈ [1,∞]. Consider A1 ∈ C (S ,GLd(R)) and denote by ( fˆ,Aˆ) the random product of (θi,Ai). Note that, for p,z ∈ X as in Section 3,

s −1 u hz,p(t) = f0 ◦ f1 and hp,z = I.

s u −1 u So, h = hz,p ◦ hp,z = f0 ◦ f1. Moreover, since t = hp,z(t) and observing that

s −1 u H(z,t)(p,h(t)) = A0(h(t)) A1(t) and H(p,t)(z,t) = I, we get that

s u −1 r 1 Ht = H(z,t)(p,h(t)) ◦ H(p,t)(z,t) = A0(h(t)) A1(t) ∈ C (S ,GLd(R)).

Therefore, to see that a random product ( fˆ,Aˆ) is twisting, we must show that for any I,J ⊂ {1,...,d} with same cardinality the map PI,J : GLd(R) → R satisﬁes

Z −1 log PI,J(A0(h(t)) A1(t)) dt < ∞. (4.3) S1 The next proposition (which will be proved in Section 4.4) is the technical −1 tool in the proof that property (4.3) above holds for A0(h(t)) A1(t) in a dense r 1 subset of C (S ,GLd(R)) and we will state it for more general polynomial maps (polynomials with the variable being the coordinates of the matrix in GLd(R)), than PI,J.

Proposition I.10. Let P : GLd(R) → R be a non-constant polynomial map, d ≥ 1 and r ∈ [1,∞] ∪ {ω}. Then, the set

r 1 1 1 AP = A ∈ C (S ,GLd(R));log|P ◦ A| ∈ L (S ,m) ,

r 1 is open and dense subset of C (S ,GLd(R)). Using the Proposition I.10 and the continuity of the invertible map r 1 r 1 φA0 : C (S ,GLd(R)) → C (S ,GLd(R)) given by

−1 φA0 (A)(t) = A0(h(t)) A(t),

25 r 1 ˆ ˆ we can see that the set of A1 ∈C (S ,GLd(R)) such that the random product ( f ,A) is twisting, that is,

−1 1 1 PI,J ◦ (A0(h(t)) A(t)) ∈ L (S ,GLd(R)),

for all maps PI,J (there are only ﬁnite of them) is open and dense. Hence, we conclude the theorem for r ∈ [1,∞] ∪ {ω}. 1 1 Assume now that r ∈ [0,1). Then, we can approximate A1 by B1 ∈ C (S , r GLd(R)) in the C topology and, after that, using the Proposition I.10 again, ﬁnd 1 r D1 close to B1 in the C topology (and then close to A1 in the C topology) such

that the random product of (θi,Di)i∈Ik is twisting, where D j = A j for all j 6= 1. This concludes the result for any r ∈ [0,∞] ∪ {ω}.

4.4 Proof of Proposition I.10

First we will give a brief review about basic real algebraic geometry. A subset l V ⊂ R is said to be an Algebraic set if there exist ﬁnitely many polynomials f1,..., fm ∈ R[X1,...,Xl] such that

m \ V = [ fi = 0]. i=1

Figure 4.2: Example of algebraic set. Cartan umbrella given by the equation z(x2 − y2) − x3 = 0.

An algebraic set V is said irreducible if, whenever V = F1 ∪ F2, with Fi algebraic sets, then V = F1 or V = F2. It is known, see [9, Theorem 2.8.3,page 50], that every algebraic set V can be written as the union of irreducible algebraic sets

26 V1,...,Vp such that Vi * ∪ j6=iVj. The sets Vi are called the irreducible components of V.

For a point x0 ∈ V, we deﬁne the Zariski tangent space of V at the point x0 as the linear space m n o T ZarV = \ x ∈ l f (x ) · x = , x0 R ;∇ i 0 0 i=1

where ∇ fi denotes the gradient vector of the polynomial fi. We say that a point x0 ∈ V is a regular point if

(T ZarV) = (T ZarV) x ∈ V , dim x0 min dim x ;

and x0 is a singular point of V if it is not regular. If V0 is an irreducible algebraic set we deﬁne the dimension of V0 as the num- Zar ber dim(Tx V) for any regular point x ∈V. For general algebraic sets V we deﬁne the dimension as

dimV = max{dim(Vi);i = 1,..., p},

where Vi are the irreducible components of V. It is important also to reinforce that, when V is an irreducible algebraic set and x ∈ V is a regular point of V, there exists a neighborhood of x in V which ∞ Zar is a C manifold and in this case Tx V is in fact the tangent space TxV of this neighborhood at x, see [9, page 66]. Let Sing(V) be the set of singular points of V (sometimes we will call this set singular part of V and its complement as regular part of V). It is also known, see [9, Proposition 3.3.14, page 69], that Sing(V) is an algebraic subset of V and

dim(Sing(V)) < dimV. (4.4)

That will be the crucial property in the proof of the Proposition I.10. For more details about algebraic and semi algebraic sets, see [9]. Consider P : GLd(R) → R a non-constant polynomial map. Fix r ∈ [1,∞] (we will deal with the case r = ω later) and consider the following algebraic set,

V 0 = [P = 0].

Observe that V 0 has dimension at most d2 − 1 (as deﬁned above) and then its regular part is a regular submanifold of GLd(R) of dimension equal to dimension of V 0. The idea to prove the Theorem 4.4 is the following: if the closed curve

t 7−→ Ht

27 does not intersects the algebraic set V0, then there is nothing to prove (the log of a positive function in a compact is integrable). But, in general that is not the case, intersection between Ht and V0 can be large. If that is the case we can use a version of Thom transversality theorem I.11 (V0 is not a regular manifold) to perturb Ht suitably and guarantee that the intersection is transversal (which is enough to get integrability that we are pursuing) as described in the Figure 4.3.

1 1 Figure 4.3: Transversality between V0 and A(S ) = {Ht; t ∈ S }.

Let V 1 = Sing(V 0) ⊂ V 0, be the singular part of V 0.

Theorem I.11 (Thom transversality theorem). Fix r ∈ [1,∞]. Let M be a manifold and N ⊂ M be a submanifold. Then, the set

r 1 {A ∈ C (S ;M); A t N}, is dense. If N is closed then the above set is also open.

Remark 3. A particular case of this theorem that we will use is when codimension r of N is large than 1. In this case, the unique way to a map A ∈ C (S1;M) to be transversal to N is

1 A(S ) ∩ N = ∅.

For a complete proof and details around this result, see [23, Corollary 4.12, page 56]. As a corollary of Theorem I.11 in our context, we have

r 1 1 1 Lemma I.12. Take A ∈ C (S ;GLd(R)) and assume that d(A(S ),V ) > 0. Then, r 1 for all U ⊂ C (S ;GLd(R)), neighborhood of A, there exists V ⊂ U , open, and a > 0 such that for all B ∈ V we have:

28 •d (B(S1),V 1) > a > 0

0 •B t V .

Observe that as a corollary of this lemma and compacity of S1, we have

1 0 #B(S ) ∩V < ∞.

In fact, we can assume that the intersection above has the same cardinality for every B ∈ V . 1 Let A and V as in Lemma I.12 and consider B ∈ V . Set {t1,...,tk} ⊂ S such 1 0 that {B(t1),...,B(tk)} = B(S ) ∩V . By transversality we have that,

0 0 (P ◦ B) (ti) = ∇P(B(ti)) · B (ti) 6= 0,∀i = 1,...,k.

1 Set s1 = 2 min{|ti −tl|;i 6= l} > 0. By Taylor’s formula we have

0 P ◦ B(t) = (P ◦ B) (ti)(t −ti) + oi(|t −ti|).

Consider s2 > 0 such that if |t −ti| < s2 for some i = 1,...,k, then

0 oi(|t −ti|) 1 0 (P ◦ B) (ti) − ≥ (P ◦ B) (ti) 6= 0. |t −ti| 2

In particular, there exists C1 > 0 such that if |t −ti| < s2, for some i = 1,...,k, we have 0 oi(|t −ti|) |log (P ◦ B) (ti) − | ≤ C1. |t −ti| 1 1 Deﬁne s = 2 min{s1,s2} and consider the intervals Ii = (ti −s,ti +s) ⊂ S . So, Z Z Z |log|P ◦ B(t)||dt = |log|P ◦ B(t)||dt + |log|P ◦ B(t)||dt 1 1 k k S S \ ∪i=1 Ii ∪i=1Ii = (I) + (II).

1 k By compacity of S \∪i=1 Ii, and the fact that the function P◦B does not vanish in this set, there exists another constant C2 > 0 such that

|log|(P ◦ B(t))|| ≤ C2,

1 k for every t ∈ S \ ∪i=1 Ii. This gives Z 1 k (I) = |log|P ◦ B(t)||dt ≤ C2m( \ ∪i= Ik) 1 k S 1 S \ ∪i=1 Ii

29 Note that by, Taylor’s Formula in each Ii, we have

0 oi(|t −ti|) log|P ◦ B(t)| = log (P ◦ B) (ti) + + log|t −ti|. |t −ti| Hence, by the choice of s, and the fact that the logarithm funtion is integrable at the origin, we conclude that

Z k Z (II) = k |log|P ◦ B(t)||dt = ∑ |log|P ◦ B(t)||dt ∪i=1Ik i=1 Ii k Z k Z 0 oi(|t −ti|) ≤ ∑ |log (P ◦ B) (ti) + |dt + ∑ |log|t −ti||dt i=1 Ii |t −ti| i=1 Ii k ≤ ∑ m(Ii)C1 +C < ∞. i=1

Therefore, for each B ∈ V we have that log|P ◦ B| ∈ L1(S1;m). To conclude the proof of the main lemma what is left to show is that the following set

r 1 1 1 A = {A ∈ C (S ;GLd(R); d(A(S ),V ) > 0}, (4.5)

r 1 is dense in C (S ;GLd(R)). This is a consequence, as we will see, of the fact that 1 the V is an algebraic variety of codimension greater or equal than one in GLd(R). Consider the following chain of algebraic sets:

m m−1 1 0 V V ... V V ,

where V i = Sing(V i−1). We know that this is a ﬁnite chain since V i is an algebraic variety of dimension strictly less than the dimension of V i−1. We can assume m that V is a regular submanifold of GLd(R) (the chain stops in this moment). Moreover, by deﬁnition, we have that each V i is a closed subset of V i−1 and i−1 i V \V is a regular submanifold of GLd(R) of codimension large than 1. m Using the remark 3 with M = GLd(R) and N = V which is a regular submanifold of GLd(R), we have that the set

r 1 1 m Am = {A ∈ C (S ;GLd(R)); A(S ) ∩V = ∅},

r 1 is dense in C (S ;GLd(R)). More generally, using the same argument, now with i i+1 M = GLd(R) and N = V \V , we will obtain that the set

r 1 1 i i+1 Ai = {A ∈ C (S ;GLd(R)); A(S ) ∩V \V = ∅},

30 r 1 is dense in C (S ;GLd(R)), for every i = 1,...,m − 1 (note that we are using strongly that the codimension is large enough). Hence, the set

m \ A = Ai, i=1

r 1 is (open) and dense in C (S ;GLd(R)). Observing that this intersection is the set A defined above in 4.5, we conclude the proof of the main lemma in the case r ∈ [1,∞]. Assume that r = ω. In this case we have that P ◦ A : S1 → R (P is a polynomial and in particular a analytic function) is an analytic function for any A ∈ ω 1 C (S ,GLd(R)). In this case, either P ◦ A is constant equal to zero or, has only finitely many zeros and those zeros have finite order in the sense that if we con- 1 sider a neighborhood of a zero ti ∈ S and write

mi P ◦ A(t) = (t −ti) gi(t),

where mi ∈ N, gi 6= 0 on this neighborhood. Since the function logt is m-integrable on [0,1], we conclude that

1 1 log|P ◦ A| ∈ L (S ,m),

ω 1 for all A ∈ C (S ,GLd(R)) such that P◦A is not zero. Assume then that P◦A ≡ 0. Since, the polynomial P is non constant, the interior of [P = 0] is empty. So, for any A ∈ [P = 0] and for any ε > 0, there exist B ∈ GLd(R), ||B|| = 1 and a positive 1 δ < ε such that A+δB ∈/ [P = 0]. In particular, taking A = A(t0), for some t0 ∈ S ω 1 and deﬁning A˜ ∈ C (S ,GLd(R)) given by

A˜(t) = A(t) + δB,

we have that A˜ is δ-Cω close to A and P ◦ A˜ does not vanish identically. Hence, we fall in the previous case and

1 1 log P ◦ A˜ ∈ L (S ,m).

This concludes the proof of the Proposition I.10.

31 Part II

Embedding small horseshoes in lower dimensions

32 5 INTRODUCTION

5.1 History and motivation

In his celebrated work on celestial mechanics, [41], Poincare,´ already mentioned that an indication of the complexity of the three bodies problem is given by the understanding of the dynamics of, what he called, bi-asymptotic solutions, which is, nowadays, known as transversal homoclinic intersections associated to a saddle type periodic point. By analyzing carefully this geometric phenomena, S. Smale, in the sixties, introduce the concept of horseshoe: a model of non trivial (not given by a finite amount of periodic orbits), chaotic, persistent, higher dimensional dynamics, which has a continuous pair of laminations globally defined and it is conjugated to a symbolic model. The conjugation with the symbolic model allow us to fully understand the dynamical behavior inside of the horseshoe (entropy, growth rate of periodic orbits, transitive orbits, among others). One of the simplest ways to add an extra difficulty is assuming that for some point of the horseshoe, for instance periodic, the stable and unstable manifolds intersect each other, not transversely (which would imply that this point were inside a new horseshoe) but tangentially (lack of hyperbolicity). That is the scope of the work of Newhouse in [34]. He proved that close to any given surface horseshoe, Λ, exhibiting a quadratic homoclinic tangency, associated with a periodic point p, and thickness larger than one there exists an open set U with persistence of tangencies. For a precise statement and the definition of thickness see [39]. In the case of surface horseshoes the stable and unstable laminations can be extended to regular (C1+) foliations. Using this regularity Newhouse was able s s to move the regular Cantor sets K (p) = Wloc(p) ∩ Λ (stable Cantor at p) and u u K (p) = Wloc(p) ∩ Λ (unstable Cantor at p) along these foliations to get regular Cantor sets on the same line (line of tangencies). When we unfold this tangency these new Cantor sets move through the line and its intersection are equivalent to new tangencies. This observation was crucial to reduce the problem to the study

33 of intersections of Cantor sets on the real line which he handled using what is currently called Newhouse Gap lemma (the thickness bound is used here). Later, in [33], Newhouse proved that if the initial tangency is associated with a dissipative periodic point, then there exists a residual set of diffeomorphisms

R∞ ⊂ U with inﬁnitely many sinks. The existence of these sinks generically is now called Newhouse phenomena. This gives an indication of how complicated the dynamics can be if we leave the region that deﬁnes the horseshoe. Newhouse, Palis and Takens, in [35], initiate an attempt to fully understand the typical behavior of the dynamics f in a region U that contains the horseshoe Λ and a region V that contains the quadratic homoclinic tangency associated with a

periodic point p ∈ Λ. They proved that if dimH(Λ) < 1, then for any smooth one

parameter family (gt)|t|≤t0 unfolding a homoclinic tangency, that is

• g0 contains a tangency in V associated with pg0 (the hyperbolic continuation of p);

• (gt)−t0

we have that the set of parameters t such that Λ˜ t, the maximal invariant in U ∪ V, is hyperbolic has total density at 0. In other words, for any smooth small

family unfolding a homoclinic tangency at V, (gt), we have that Λ˜ g is hyperbolic with total density at the ﬁrst parameter. In order to prove this result Newhouse, Palis and Takens made use again of regularity of the invariant foliations to reduce the problem to study translations (arithmetic difference) of regular Cantor sets in the tangent line and whose the sum of its Hausdorff dimensions are smaller than one. In particular, the arithmetic difference between these Cantor sets has zero Lebesgue measure which indicates the density property attained by this result. The Newhouse, Palis and Takens result, above described, led the theory to a

natural question: what happens if dimH(Λ) > 1. Aiming to answer this question Palis conjectured, in [38], that for typical pairs

of Cantor sets K1 and K2 with dimH(K1)+dimH(K2) > 1 the arithmetic difference K1 − K2 contains an interval. Moreira introduced, in [31], the notion of stable intersection between the pairs k of C regular Cantor sets (K1,K2) which means that there exists a neighborhood of the pair (K1,K2) such that the intersection happens for any pair of regular Cantor sets in this neighborhood. Moreira and Yoccoz, in [32], proved that there exists an k open and dense set U of pairs of C regular Cantor sets such that if (K1,K2) ∈ U , then there exists t ∈ R such that (K1,K2 +t) has stable intersection. In their work, Moreira and Yoccoz, introduced new perturbations techniques and used tools from

34 the probability theory to address the problem. These ideas and techniques was the fuel necessary to understand the typical dynamical behavior in the region U ∪V, described above, when we unfold a quadratic tangency associated with a periodic point in a fat surface horseshoe Λ. More precisely, in [30], Moreira and Yoccoz proved that given a surface horseshoe Λ for a diffeomorphism f with dimH(Λ) > 1 and a quadratic homoclinic tangency associated with a periodic point p ∈ Λ, then

for an open and dense set U0, g0 ∈ U0, in a neighborhood of f and for all smooth one parameter family passing through g0 we have that the set of t such that gt has tangency robustly has positive inferior density at 0. Among others, the lack of regularity of the invariant foliations is one of the most fundamental difficulties when we try to push the above results from surfaces to higher dimensional horseshoes. It is not hard to build 3 dimensional horseshoes given by a transversal homoclinic intersection associated to a periodic point in which the unstable lamination is not even Lipschitz. It is enough, for example, to choose the eigenvalues of the periodic point suitably and assume that the homoclinic intersection occurs exactly over the strong stable manifold (not a generic condition). See [40]. Focusing on generalizing Newhouse results, Palis and Viana, in [40], dealt with these difficulties proving that typical transversal homoclinic intersections are inside of horseshoes with intrinsically C1+ unstable (or stable) laminations 1 which is enough to recover many of the geometric properties of the horseshoe (for example the thickness can be defined and we can preserve regularity of regular Cantor sets by holonomies). Using this weak notion of regularity, Palis and Viana, were able to prove that near any smooth diffeomorphism exhibiting a tangency associated with a sectional dissipative saddle, there exists a residual subset of an open set such that each of its elements displays infinitely many sinks. Another problem in the study of higher dimension horseshoes is the bad behavior of the Hausdorff dimension of the horseshoe with respect to the diffeomorphism. In [13], Bonatti, Diaz and Viana build an example of a horseshoe Λ, for a diffeomorphism f , in dimension 3, with dimH(Λ) < 1 which can be approxi- mated by diffeomorphisms g such that dimH(Λg) > 1 (Λg is the hyperbolic continuation of Λ, see Section 6.2). The idea is again to consider a horseshoe Λ, with

1Intrinsic derivatives (also known as Whitney’s derivatives) is a notion that can be deﬁned, differently of the traditional derivatives, on compact sets. Whitney proved that when a function is C1 intrinsic, there exists an open set such that the function can be extended to this open set, but in general this extension does not give any information for the point of view of the original dynamics.

35 dimH(Λ) < 1, given by a transversal homoclinic intersection happening exactly, again, over the strong stable leaf of the periodic point (not generic). When this intersection is unfolded it emerges horseshoes with Hausdorff dimension larger than one. In [29], Moreira, Palis and Viana, propose a script to generalize the New- house, Palis and Takens result and Moreira and Yoccoz result to higher dimensional horseshoes. They suggest to use a different notion of dimension, the upper stable dimension, inspired by the concept of afﬁne dimension introduced by Fal- coner in [20], which seems to suit better from a dynamics point of view.

The upper stable dimension, d¯s(Λ), which measures the size of the hyperbolic set inside the stable leaves, is always upper semi continuous with respect to the diffeomorphism (under extra hypothesis is continuous, see Chapter 8) and is an s ¯ upper bound for stable dimension at any x ∈ Λ, that is, dimH(Wloc(x)∩Λ) ≤ ds(Λ) which motivates the upper in the name of d¯s(Λ). See Section 7.2 to a precise deﬁnition of the upper stable dimension. In [28], Matheus, Moreira and Silva used the upper stable dimension to study horseshoes with sharp s-splitting (see Section 6.1). They proved that typical C∞ large (stable dimension larger than one for any point in the horseshoe) horseshoes contains blenders of codimension one. For more details about blender see [12]. Among other things, the present work has the intention to continue the formal- ization of the ideas described in the script proposed by Moreira, Palis and Viana. In the next section, we describe our results.

5.2 Results

The examples mentioned in the previous section gives an indication that strong stable leaves passing through many points of a given horseshoe is an obstruction to good geometric properties of the horseshoe. The main result of this thesis is a ﬁrst attempt to deal with this kind of issue.

Theorem A (Main Theorem). Let M be a compact manifold, dim(M) ≥ 2, and r r > 1. Consider Λ ⊂ M be a horseshoe for f ∈ Diff (M) with a s-splitting, TΛM = ss w u w E ⊕ E ⊕ E . Assume that dim(E ) = 1 and d¯s(Λ) < 1. Then, given ε > 0 r r 0 there exists g ∈ Diff (M),C close to f , and a subhorseshoe Λg of the hyperbolic continuation of Λ for g, Λg, such that

¯ 0 ¯ 1. ds(Λg) ≥ ds(Λg) − ε;

36 0 2. For every x ∈ Λg we have

ss 0 W (x) ∩ Λg = {x}.

Theorem A has the following geometric property which is a direct consequence of the characterization obtained by Bonatti and Crovisier in [11].

Theorem B (Dimension reduction). There exists a locally g-invariant, C1+, sub- 0 0 w manifold S of M which contains Λg and such that for every x ∈ Λg,TxS = E (x)⊕ Eu(x).

0 Figure 5.1: Λg embedded in S

5.3 Work in progress

We want to point out that, in typical terms, the hypothesis of the Theorem A r are not particular in the sense that for a typical C horseshoe Λ with d¯s(Λ) < 1 there exists a subhorseshoe with almost the same upper stable dimension which has s-splitting. This is attained ﬁrst guarateeing the existence of a periodic point

37 with simple spectrum (typical among the Cr horseshoe) and later building random perturbations (see Section 9.1) to suitably find cone fields which will led us to a dominated splitting inside the stable bundle. An interesting consequence of the Theorem A is the regularity (C1+) of the 0 unstable laminations of the subhorseshoe Λg meaning that this lamination can be extended to a C1+ foliations (similarly to what happens in the case of surface 0 horseshoes). The reason is that since the strong stable leaves only intersect Λg in a single point (Ess is a normal direction), we can ignore the strong stable direction 0 and so guarantee that bunching conditions are satisfied for Λg. Using, therefore, the theory developed by Hirsch, Pugh and Shub [25] we will be able to obtain the aimed regularity. Adding the information provided above we will achieve the following result:

Proposal 1 (In progress). For typical Cr diffeomorphism f with horseshoe Λ with dimH(Λ) > 1 we can build subhorseshoes Λ1 and Λ2 which are invariant by some iterate f k satisfying that:

1. Λ1 has a globally deﬁned strong stable lamination and regular unstable holonomies;

2. Λ2 has a globally deﬁned strong unstable lamination and regular stable holonomies;

3. It holds

max < 1 and d¯s(Λ1) + d¯u(Λ2) > 1. d¯s(Λ1), d¯u(Λ2)

4. For each p ∈ Λ and p ∈ Λ the projections πss| s , along the 1 1 2 2 Λ1∩Wloc(p1) strong stable leaves, and πuu| u , along the strong unstable leaves, Λ2∩Wloc(p2) are injective and its inverse has a C1+ extension.

38 6 HORSESHOES

6.1 Dominated splitting and Partially hyperbolic sets

Let Λ ⊆ M be a compact manifold. A splitting of TΛM is a decomposition

TΛM = E1 ⊕ ··· ⊕ Ek, in continuous linear ﬁber subbundles Ei ⊆ TM over Λ, where the dimension of Ei(x) is independent of x ∈ Λ, for all 1 ≤ i ≤ k. Let f ∈ Diffr(M), r ≥ 1, and assume that Λ ⊆ M is a compact f -invariant set.

We say that a splitting TΛM = E1 ⊕ ··· ⊕ Ek is a dominated splitting for Λ if it satisﬁes

• Invariance: d f (x)Ei(x) = Ei( f (x)), for every x ∈ Λ and for 1 ≤ i ≤ k;

• Domination: There exist C > 0 and σ < 1 such that

n n n d f (x)|Ei(x) ≤ Cσ m(d f (x)|Ei+1(x))

for all n ≥ 1 and all 1 ≤ i ≤ k − 1.

Λ is said to be a partially hyperbolic set for f if admits a dominated splitting

TΛM = E1 ⊕···⊕Ek (partially hyperbolic splitting) such that E1 is uniformly contracted and Ek is uniformly expanded, that is, there exist C > 0 and λ ∈ (0,1) such that n n −n n d f (x)|E1(x) ≤ Cλ d f (x)|Ek(x) ≤ Cλ . c s u In this case we call the bundles E2 ⊕ ··· ⊕ Ek−1 = E , E1 = E and Ek = E , respectively, the center, stable and unstable bundles for f over Λ. When the partially hyperbolic set Λ has trivial center bundle, Ec = {0} we say that Λ is a (uniformly) hyperbolic set for f ∈ Diffr(M). ss w u When Λ is hyperbolic with a (nontrivial) splitting TΛM = E ⊕E ⊕E , with Es = Ess ⊕ Ew uniformly contracted and Eu uniformly expanded, the bundles Ess

39 and Ew are called respectively strong stable bundle and weak stable bundle and ss w u we say that Λ has a s-splitting TΛM = E ⊕E ⊕E . Analogously, we can deﬁne a u-splitting for Λ. Observe that this notion requires that the dimension of the manifold M must to be at least 3. A riemannian metric on M is said to be adapted to a partially hyperbolic set s c u Λ, for f , if the partially hyperbolic splitting TΛM = E ⊕ E ⊕ E satisﬁes that:

• d f (x)|Es(x) < 1,

−1 • d f (x)|Eu(x) < 1 and

• d f (x)|Es(x) ≤ m(d f (x)|Ec(x)) ≤ d f (x)|Ec(x) ≤ m(d f (x)|Eu(x)).

Basically, the existence of such a metric on the manifold M indicates that the contraction, expansion and the domination in the partially hyperbolic splitting of

TΛM can be see in the ﬁrst iteration of the dynamics f . It is known that given a partially hyperbolic set for f ∈ Diffr(M) in a compact manifold M with riemannian metric g, there exists a riemannian metricg ˜, equivalent to g, which is adapted to Λ. For more details see ([16], page 17). s c u We say that a splitting, TΛM = E ⊕ E ⊕ E , of a partially hyperbolic set Λ (with an adapted metric) is a sharp splitting if there exist positive real numbers λ sc and λ cu such that

sc sup d f (x)|Es(x) < λ < inf m(d f (x)|Ec(x)) x∈Λ x∈Λ cu ≤ sup d f (x)|Ec(x) < λ < m(d f (x)|Eu(x)). x∈Λ Similarly we can deﬁne sharp dominated splitting when we have a dominated splitting instead of a partially hyperbolic splitting of the set Λ. ss w u In the case that Λ is hyperbolic with a s-splitting, TΛM = E ⊕ E ⊕ E , we can interpret Λ as a partially hyperbolic set with nontrivial center bundle Ec = Ew.

We say that Λ has a sharp s-splitting if the (partially hyperbolic) splitting TΛM = Ess ⊕ Ew ⊕ Eu is a sharp splitting. In a similar way we deﬁne a sharp u-splitting for Λ. All the concepts introduced in this section are robust which means that if, Λ ⊆ M is a compact set satisfying the P property, where P stands to either

1. Λ has a dominated splitting or

2. Λ has a partially hyperbolic splitting or

40 3. Λ has a hyperbolic splitting, then there exist an open neighborhood U ( f ) ⊂ Diff1(M) of f ∈ Diffr(M) and an open neighborhood U of Λ such that if g ∈ U ( f ) and K ⊂ U is a g-invariant set, then K satisﬁes the property P (see [16], page 16).

6.2 Horseshoes

Let Λ ⊆ M be a (uniformly) hyperbolic set for f ∈ Diffr(M), r ≥ 1, with an adapted metric. We will assume throughout this entire work that our metric is adapted, unless we make explicit the contrary. Λ is said to be isolated if there exists open neighborhood, U ⊆ M, of Λ such that \ Λ = f j(U). j∈Z Deﬁnition II.1. A compact set Λ ⊆ M is called a horseshoe for a Cr diffeomorphism, f : M → M, if it is an inﬁnite, isolated, topologically mixing and totally disconnected hyperbolic set.

We say that Λ is a Cr horseshoe if Λ is a horseshoe for some f ∈ Diffr(M).

Example II.2. Let f : R2 → R2 which maps the unit square Q ⊂ R2 in the ”horseshoe shape” as described in the ﬁgure 6.1. The maximal invariant inside the square Q is called Smale horseshoe.

Figure 6.1: Smale horseshoe.

Example II.3. In the same spirit of the dynamics of the Smale horseshoe we can deﬁne horseshoes by homoclinic intersections as described in the ﬁgure 6.2.

41 Figure 6.2: Homoclinic intersection in dimension 2.

Example II.4. In the case that the periodic point, with transversal homoclinic intersection, has simple spectrum we can build a horseshoe with s-splitting.

Figure 6.3: s-splitting.

Example II.5. The case of complex eigenvalues led to the a horseshoe without s-splitting.

Remark 4. If Λ is a horseshoe, then Per( f |Λ) = Λ. That can be proved using the shadowing property of hyperbolic sets (or the conjugation with the shift map (see Section 6.5).

Remark 5. The isolated condition for a hyperbolic set Λ is equivalent to the Local product structure which means that the intersections of stable and unstable manifolds of points of Λ lies in Λ (see Section 6.3).

42 Figure 6.4: Non s-splitting.

Let Λ be a horseshoe for a f ∈ Diffr(M) and consider U ⊆ M be a small open neighborhood of Λ. As mentioned in the end of the Section 6.1, choosing U small enough, we have that for any g ∈ Diff1(M) close to f in the C1 topology, if K ⊆ U is a g-invariant set, then K is hyperbolic. Since Λ is a horseshoe, a stronger conclusion holds. There exists a neighborhood U ( f ) in Diff1(M) and a 0 continuous map φ : g ∈ U ( f ) 7−→ φg ∈ C (Λ;M) satisfying that

1. There exists K > 0 such that

dC0 (incΛ,φg) ≤ KdC0 ( f ,g) (6.1)

0 where incΛ ∈ C (Λ;M) is the inclusion map;

2. The following diagram

f Λ / Λ

φg φg Λg g / Λg

commutes.

See ([46], page 101). The sets Λg = ϕg(Λ) are called hyperbolic (or analytical) continuations of Λ by g. They satisﬁes that

\ n Λg = g (U), n∈Z r r for any g ∈ U ( f ). In particular, Λg are C horseshoes for all g ∈ U ( f )∩Diff (M).

43 6.3 Invariant manifolds

In what concerns the study of regularity of invariant foliations by diffeomorphisms there are three different (although intrinsically related) classes of problems.

1. Existence, uniqueness and regularity of the leaves;

2. Regularity of the foliations/lamination (transversal regularity or regularity by holonomies);

3. Regularity of the leaves with respect to the hyperbolic continuation of the leaf (regularity with respect to the diffeomorphism).

In this section we will discuss and try to bring references about each of these problems. Consider Λ ⊂ M a horseshoe for a diffeomorphism f ∈ Diffr(M) with s- ss w u splitting, TΛM = E ⊕ E ⊕ E . s The stable manifold theorem states that there exists a continuous map Φ f : r Es Λ → Emb (Ddim ;M) such that

s s Φ f (x) = Wf ,ε (x) for some ε > 0. This implies that there exists a unique stable lamination

s s F f = {Wf (x);x ∈ Λ},

s of Λ by stable manifolds such that each leave Wf (x) is an immersed submanifold of M of class Cr. There also exists a continuous map

1 s r dimEs g ∈ U( f ) ⊂ Diff (M) 7−→ Φg ◦ φg : Λ → Emb (D ;M)

The same can be said about the unstable lamination. So,

s u r (i) Wf ,Wf always exist, are unique and have C leaves;

s (ii) The maps g ∈ U( f ) 7−→ Wg,ε (φg(x)) are continuous for all x ∈ Λ. See for example [39]. We have similar results for the strong stable foliation/lamination F ss (the one whose leaves are tangent to the strong stable space Ess). Deﬁne

ss n n −εn n W (x) = y ∈ M; d( f (x), f (y)) ≤ Ce m(d f (x)|Ew(x)) .

44 This deﬁnition does not depends on the choice of (small) ε > 0. This lamination

for x ∈ Λ is also unique (if the splitting of TΛM is ﬁxed which our case is). (see [16]). In [25] is proved that for every x ∈ Λ, W ss(x) is uniquely deﬁned, invariant by r ss f and is a C -immersed submanifold of M tangent to E . For η > 0, if Dη (x) ⊂ W ss(x) is a disc of length η in W ss(x) it depends continuously on f in the Cr- topology and x ∈ Λ. So,

(1) W ss always exists, is unique and have Cr leaves.

ss (2) The map g 7−→ Wg,η vary continuously.

At this moment we wonder if there exists a weak stable lamination tangent to Ew. If we assume that Λ has a sharp s-splitting we can use the theorem in ([39], page 158) to prove that we have a weak unstable manifold tangent to Ew ⊕ Es deﬁned for any x ∈ Λ and intersect this manifold with W s to guarantee the existence of W w. But, that was only possible since we have a sharp separation between the bundles Ess and Ew ⊕ Eu. In general, the existence of the weak stable manifold can fail and even when exists the unicity can be not assured. See [39]. Until now the discussion is pretty much satisfactory to address the classes of problems mentioned in the beginning of section in a continuous fashion. What can we say about higher regularities? If r > 1, then the bundles Es,Eu,Ess and Ew are always Holder¨ continuous ([16], page 44).

Remark 6. If f ∈ Diff1(M), then in general the bundles Es and Eu are only continuous. It is possible to build examples of 2-dimensional Anosov diffeomorphisms which are C1 but the bundle are only continuous using Franks lemma [21].

In [44] they proved that W s,W ss and W u are also Holder¨ continuous. In [45] they give a better explanation of the fact that inside of a leaf of the stable lamination, W s(p), the lamination W ss can be extended to a regular C1+ foliation. This last fact will repeatedly be used throughout this work. r Let { fµ }µ∈X be a C family of diffeomorphisms. For µ0 ∈ X, let Λµ0 be a

horseshoe for fµ0 and denote by Λµ the hyperbolic continuation of Λµ0 for any µ ∈ X (the family { fµ } is taken in such a way that the hyperbolic continuation

is well deﬁned). Take x0 ∈ Λµ0 and let xµ = φ fµ (x0) ∈ Λµ be the hyperbolic

45 continuation of x0. The map µ 7−→ xµ , is Cr (See Appendix 1 in [39]). This fact will be useful in the subsequent sections. Summarizing the discussion so far:

s u ss 1. There are uniquely deﬁned laminations (continuous), W f , W f and W f such that the leaves are of class Cr which is the regularity of f ;

2. These laminations are Holder¨ continuous at Λ, W ss vary in a C1+ fashion inside of any leaf of the stable lamination W s.

3. The existence of a weak stable lamination, W w, tangent to Ew can fail and even in the cases in which such a lamination exists there is no reason to be unique (in general it is not);

∗ 4. The maps g ∈ U ( f ) 7−→ Wg,ε (φg(x)) are only continuous (actually, we have r dimE∗ the continuity of the map g ∈ U ( f ) 7−→ Φg ◦ φg : Λ → Emb (D ;M) ∗ dimE∗ where Wg,ε (φg(x)) = Φg(φg(x))(D ). r In any dimension, we have that the map g ∈ U ( f ) ⊂ Diff (M) 7−→ φg(x) is of class Cr for any x ∈ Λ. Here U ( f ) is endowed with the Cr topology.

Remark 7. More can be said in the two dimensional case where we have a better understanding of the regularity of these laminations (which can be extended to C1+ foliations in a neighborhood of Λ) and hence of the geometric properties of surface horseshoes.

Remark 8. Another possible (future) discussion is regarding the regularity of the holonomies. The Holder¨ regularity (transversal regularity) of the foliations described in this section stands for the Holder¨ regularity of the holonomies as well. For a more general discussion about the different types of regularity of general foliations see [44].

6.4 Markov partitions

Let Λ be a horseshoe for f ∈ Diffr(M). A subset P of Λ is said to be a rectangle

if for any x,y ∈ P, [x,y]ε0 ∈ P, where

{[x,y] } = W s (x) ∩W u (y). ε0 ε0 ε0

We say that a rectangle P is proper if P = IntΛ P.

46 s Let P be a rectangle with diam(P) ≤ ε0. For any x ∈ P, denote by W (x;P) the W s (x) ∩ P set ε0 . A Markov partition for a horseshoe Λ, for f ∈ Diffr(M), is a ﬁnite partition P = {P(a);a ∈ A} formed by proper rectangles of Λ satisfying:

(i) IntΛ P(a) ∩ IntΛ P(b) = ∅ for all a,b ∈ A;

−1 (ii) For any a,b ∈ A and x ∈ Λ such that x ∈ IntΛ P(a)∩ f (IntΛ P(b)) we have

f (W s(x;P(a))) ⊂ W s( f (x);P(b)) f (W u(x;P(a))) ⊃ W u( f (x);P(b)).

The set A is called the alphabet associated to the Markov partition P. The elements of P will frequently be called by either boxes or pieces of the Markov partition. It was proved by Bowen, in [15], that given a horseshoe Λ we can ﬁnd Markov partitions with arbitrarily small diameters. We will use these information in the subsequent sections without make any mention.

Figure 6.5: Markov Partition with small diameter.

6.5 Symbolic dynamics

Given a horseshoe Λ for f ∈ Diffr(M) with Markov partition P = {P(a);a ∈ A}, we say that a pair (a,b) ∈ A2 is admissible if f (P(a)) ∩ P(b) 6= ∅. Deﬁne n o Z Σ = θ ∈ A ; (θi,θi+1) is admissible for everyi ∈ Z .

Σ is a compact subset of AZ with the product topology and σ : Σ → Σ (the shift map) is a homeomorphism.

47 In [15], it was also prove that there exists a homeomorphism i f : Σ → Λ which make the following diagram

σ Σ / Σ

i f i f Λ / Λ f

commutes. The pair (σ,Σ) is called symbolic representation of the horseshoe

dynamics ( f |Λ,Λ). For each a ∈ A consider an small open neighborhood U(a) of P(a) such that U(a) ∩U(b) = ∅, for all a,b ∈ A (this is possible because |A| < ∞ and the piece P(a) are two by two disjoints and so have positive distance). Set U = ∪a∈AU(a) and let U ( f ) ⊂ Diffr(M) be an open neighborhood of f such that for any g ∈ U ( f ) the analytical continuation of Λ by g, Λg, is contained in U. The set Pg = Pg(a) := U(a) ∩ Λg;a ∈ A

is a Markov partition for Λg and there exists ig : Σ → Λg homeomorphism such that the following diagram

σ Σ / Σ

ig ig Λg g / Λg

commutes for all g ∈ U ( f ). For each element x ∈ Λg, there exist θ = (θi)i∈Z ∈ Σ such that \ − j {x} = g (U(θi)). j∈Z

6.6 Symbolic dynamics notation

Now we introduce a notation which will be used throughout this work.

• Positive inﬁnite sequences or (local) stable manifolds: n o + + + N0 + + Σ = θ = (θn )n≥0 ∈ A ; θi θi+1 is admissible ;

• Negative inﬁnite sequences or (local) unstable manifolds: n o − − − N0 − − Σ = θ = (θ−n)n≥0 ∈ A ; θ−iθ−i+1 is admissible ;

48 • Positive ﬁnite words or stable cylinders:

+ W = {a = a0a1 ...an; n ∈ N0, ai ∈ A and aiai+1 is admissible};

• Negative ﬁnite words or unstable cylinders:

− W = {a = a−n ...a−1a0; n ∈ N0, a−i ∈ A and a−ia−i+1 is admissible};

• (Bi) ﬁnite words or boxes:

W = {a = a−n ...a0 ...am; n,m ∈ N0, ai ∈ A and aiai+1 is admissible};

We see all the above introduced sets Σ+,Σ−,W+,W−,W as subsets of 2Σ, meaning that any element θ (or a) in this sets is a subset of Σ. For example an element a = a−n ...a0 ...am ∈ W is the subset of elements θ ∈ Σ such that θi = ai for every −n ≤ i ≤ m. We also note that W+ ⊂ W and W− ⊂ W, once that if + − a = a−n ...a0 ...am ∈ W with n = 0 we have a ∈ W and if m = 0, a ∈ W . + − If a = a0 ...am ∈ W and b = b−k ...b0 ∈ W with a0 = b0, we use the notation

b ∩ a = b−k ...b0a1 ...am ∈ W, to the actual intersection between the subsets a and b of Σ. Using this notation we can always write any element a ∈ W as a = a− ∩a+, with a− ∈ W− and a+ ∈ W+. We set + : Σ → Σ+ and − : Σ → Σ− the natural projections

+(θ) = θ + − (θ) = θ −.

So, giving any θ ∈ Σ, there exist θ + ∈ Σ+ and θ − ∈ Σ− such that θ = θ − ∩ θ + (this is an abuse of the language since, θ is an element of Σ and θ − ∩ θ + is a subset of Σ which is equal to {θ}). Observe that the shift map naturally acts in these sets: If a ∈ W, for any j ∈ Z,

σ j(a) = θ ∈ Σ;σ − j(θ) ∈ a and if θ ± ∈ Σ±, then

σ j(θ ±) = θ ∈ Σ;σ − j(θ) ∈ θ ± .

It is worthwhile to mention that when a = a−n ...a0 ...am ∈ W and −n ≤ j ≤ m, we still see the set σ j(a) as an element of W that, although has the same symbols as a, is a different element in W (the position of the 0-th coordinated

49 changes). In other words, if, without loss of generality, j ≥ 0, σ j(a) = b = b−n− j ...b0 ...bm− j ∈ W, where bi− j = ai for −n ≤ i ≤ m. For any a = a−n ...a0 ...am ∈ W, |a| = m + n + 1 denote the size of the word a. Consider a = a−n ...a0 ...am ∈ W and b = b−k ...b0 ...bl ∈ W ﬁnite words 2 such that the pair (am,b−k) ∈ A is admissible. Deﬁne the right concatenation of the word a by b as the word

a · b = c = c−n ...c0 ...cm+|b|W R with ( ai if − n ≤ i ≤ m ci = bi−m−1−k if m + 1 ≤ i ≤ m + |b| We deﬁne the left concatenation of the word b by the word a as the word

a · b = d = d−k−|a| ...d0 ...dl ∈ W, L with ( bi if − k ≤ i ≤ l di = ai+k+1+m if − k − |a| ≤ i ≤ −k − 1 Observe that these operations are not commutative, i.e., in general, we do not have a · b 6= b · a and a · b 6= b · a. R R L L We denote by a·n the right concatenation of the word a ∈ W by a, n-times. R Analogously we define ·nb the left concatenation of the word bW by b n-times. L Since W+ ⊂ W the definition of right concatenation applies to this finite words but in this case is more natural to denote a · b only by ab ∈ W+. The same R holds to the left concatenation between negative finite words, c,d ∈ W−, which will be denoted by cd = c · d ∈ W−. In this case for a ∈ W+,b ∈ W− we write L a·n = an and ·nb = bn. Remember that this concatenations are only possible if the R L admissible properties are satisfied by the words a and b. + Given a = a0 ...am,b = b0 ...bl ∈ W with a0 = b0 we define the maximum common word in a and b as the word

+ a ∨ b = a0 ...ak ∈ W , where a j = b j for all 0 ≤ j ≤ k and ak+1 6= bk+1. Similarly, given c = c−n ...c0,d − = d−k ...d0 ∈ W , with d0 = c0 we deﬁne the minimum common words in c and d as the word − c ∧ d = d−m ...d0 ∈ W ,

50 where ci = di for all −m ≤ i ≤ 0 and c−m−1 6= d−m−1. Note these deﬁne commutative operations: a ∨ b = b ∨ a and c ∧ d = d ∧ c. This above introduced symbolic notations and operations turns out to be really useful when we are dealing with horseshoes. A ﬂavor of such applications can be seen when we consider the following set of notations. Let f ∈ Diffr(M) with a horseshoes Λ which is conjugated to Σ. As described in the end of the Section 6.2 there exists a neighborhood U ( f ) ⊆ Diff1(M) such that the hyperbolic continuation of Λ for any g ∈ U ( f ), Λg is also conjugated to Σ with continuous conjugation ig : Σ → Λg. For a = a−n ...a0 ...am ∈ W and g ∈ U ( f ), write

n \ − j Pg(a) = g (Pg(a j)) ⊂ Λg. j=−m

This set is called a piece (or box) associated to a in the horseshoes Λ. When + a ∈ W the set Pg(a) is denoted by Hg(a) and is called stable cylinder and when − a ∈ W , the set Pg(b) is denoted by Vg(a) and is called unstable cylinder. Set, for θ + ∈ Σ+ the local stable manifold associated with θ + as

s + \ − j + Wg,loc(θ ) = g (U(θ j )), j∈N and for θ − ∈ Σ−, the local unstable manifold associated with θ − as

u − \ j − Wg,loc(θ ) = g (U(θ j )). j∈N These are open subsets of some leaf in the stable and unstable lamination of Λ respectively. Note that, by the conjugation property, if a ∈ W−, we have

−|a| −|a| g (Vg(a)) = Hg(σ (a)).

51 Figure 6.6: From stable cylinder to unstable cylinder.

Another important observation is that the restriction of ig to a, ig|a : a → Pg(a) ± s/u ± is also a homeomorphism (the same holds for θ and Wg,loc(θ ) instead of a and Pg(a)). + − The homeomorphisms, i|∗, ∗ ∈ {a,θ ,θ }, allow us to identify boxes, unstable/stable cylinders and local stable/unstable manifolds with its symbolic repre- sentations.

52 7 METRIC INTERACTIONS BETWEEN HORSESHOES AND ITS SYMBOLIC MODELS

We start this section describing the metrics deﬁned on Σ, Σ− and Σ+ which will be used in all the metric consideration that we will make in this work (remember the notation introduced at Section 6.6). Deﬁne, for θ +,θ˜ + ∈ Σ+

+ + −|θ +∨θ˜ +| d+(θ ,θ˜ ) = 2 , and for θ −,θ˜ − ∈ Σ− − − −|θ −∧θ˜ −| d−(θ ,θ˜ ) = 2 . Taking θ,θ˜ ∈ Σ and writing θ = θ − ∩ θ + and θ˜ = θ˜ − ∩ θ˜ + with θ −,θ˜ − ∈ Σ− and θ +,θ˜ + ∈ Σ+ set

− − + + d(θ,θ) = max d−(θ ,θ˜ ), d+(θ ,θ˜ ) .

Using the standard metric introduced above we can prove (see [26], page 598) that the continuous conjugation i f : Σ → Λ actually is a bi-Holder¨ map (but in general not Lipschitz) with uniformly Holder¨ exponent and Holder¨ constant in a neighborhood U ( f ) ⊂ Diffr(M), this will be useful to transcribe metric properties from symbolic dynamics to the horseshoes which can be exempliﬁed by the next proposition whose the proof will be skipped (use the fact that the conjugation i f is Holder¨ continuous to pass the problem to symbolic dynamics).

Proposition II.6. Let P be a Markov partition for Λ with alphabet A. For any g ∈ U ( f ), if diam(Pg(a)) ≥ ζ > 0 for some a ∈ W, then

1/δ dist(Λ\Pg(a);Pg(a)) & ζ 0

− −1 −1 where δ0 = [i f ][i f 1] and [i f ],[i f ] ∈ (0,1] are the Holder¨ exponents of i f and i f respectively.

53 Figure 7.1: Piece P(a) with diameter larger than ζ.

7.1 Stable diameters

Let Λ be a horseshoe for f ∈ Diffr(M) and P be a Markov partition with alphabet A for Λ. For each b ∈ W−, deﬁne

|b| Ds( f ;b) = sup d f (x)|Es(x) . −|b| x∈ f (Vf (b))

Ds is called the stable diameter along Vf (b). When the diffeomorphism f is ﬁxed we will only write Ds(b) instead of Ds( f ;b). Set

−1 −1 τ = sup dg (x)| s and τˆ = sup dg(x)| s . g Eg(x) g Eg(x) x∈Λ x∈Λ

For any g ∈ U ( f ). Note that 0 < τg ≤ τˆg < 1 and that if U ( f ) is chose small enough, there exists τ,τˆ ∈ (0,1) such that

0 < τ ≤ τg ≤ τˆg ≤ τˆ < 1, (7.1) for every g ∈ U ( f ).

Proposition II.7. 1. For any b ∈ W−,

|b| |b| τ f ≤ Ds(b) ≤ τˆf ;

2. If b,c ∈ W− we have

Ds(bc) ≤ Ds(b)Ds(c).

54 Proof. Note that for every x ∈ Λ and for any n ∈ N

n n d f (x)|Es(x) ≤ τˆf and −n −n d f (x)|Es(x) ≤ τ f . So, n n n n τ f ≤ m(d f (x))|Es(x) ≤ d f (x)|Es(x) ≤ τˆf , for every x ∈ Λ and n ∈ N. Taking n = |b| we conclude the item 1. −|bc| −|b| |b| Observe that if x ∈ f (Vf (bc)), then x ∈ f (Vf (b)) and f (x) −|c| ∈ f (Vf (c)) and

|bc| |c| |b| |b| d f (x)|Es(x) ≤ d f ( f (x))|Es( f |b|(x)) · d f (x)|Es(x) , which implies

|bc| d f (x)|Es(x) ≤ Ds(b) · Ds(c), −|bc| for every x ∈ f (Vf (bc)). So,

Ds(bc) ≤ Ds(b) · Ds(c).

For b ∈ W− set

Dˆ (b) = W s (y) ∩V (b) , s sup diam ε0 f y∈Λ∩Vf (b) where the diameter used is deﬁned using the metric in the manifold M.

Figure 7.2: Stable diameter

55 − −|b| s Consider b ∈ W and x1,x2 ∈ f (Vf (b)) such that x2 ∈ W (x1) and

d( f |b|(x ),d f |b|(x )) = W s (y) ∩V (a), 1 2 diam ε0 f

|b| where y ∈ f (x1). By mean value theorem,

s |b| diam Wε (y) ∩Vf (b) ≤ sup d f (z)|Es(z) · d(x1,x2) −|b| z∈ f (Vf (b))

≤ diam(Λ) · Ds(b).

Since x1 and x2 have been choose freely we can take the supremmum in y ∈ Vf (b) in both sides of the above inequality to conclude the proof of the following proposition.

Proposition II.8. For any b ∈ W−

Dˆ s(b) ≤ diam(Λ) · Ds(b).

Remark 9. In general, we also can guarantee that for any b ∈ W−

|a| ˆ inf m(d f (z)|Es(z)) . Ds(b). z∈Vf (b) However in the Section 8.1 we will see that in the case of a one dimensional weak stable bundle, Ds(b) and Dˆ s(b) are comparable.

7.2 Upper dimensions

Let Λ be a horseshoe for f ∈ Diffr(M) and P be a Markov partition with alphabet A for Λ. The goal of this section is to introduce the notion of the upper stable dimension for the horseshoes. This dimension has the advantage of using the potentialities of dynamically deﬁned sets and seems to be more manageable than the Hausdorff dimension in the case of higher dimension (larger than 3) horseshoes.

Consider the function γn : [0,∞) → [0,∞) which is given by

t γn(t) = ∑ Ds(b) . b∈W−:|b|=n

The next proposition list some of the properties of the function γ. The proof is simple and will be skipped.

Proposition II.9. It holds:

56 (i) γn(0) = |{b ∈ W; |b| = n}| > 1;

(ii) γn(∞) = 0;

(iii) γn is strictly decreasing and convex for every n ≥ 1;

(iv) (γn) is a sequence of functions which is sub-multiplicative, i.e.,

γn+m(t) ≤ γn(t)γm(t).

Observe that the above proposition guarantees that there exists a number

λn( f ) ∈ (0,∞) which satisﬁes:

γn(λn( f )) = 1.

Fix k ∈ N and consider a large n ∈ N. Write n = qk + r where q,r ∈ N0 and 0 ≤ r ≤ k − 1. So, by the property (iv), we have

q γn(λk( f ) + ε) ≤ γr(λk( f ) + ε) · γk(λk( f ) + ε) where ε > 0. Since γk is a strictly decreasing function,

γk(λk( f ) + ε) < 1.

We also have there exists C > 0 (independent of n) such that

max {γr(λk( f ) + ε)} < C, 0≤r≤k−1 since λk( f ) + ε is close to λk( f ) for every ε small enough. Then, taking n large enough, we guarantee that

γn(λk( f ) + ε) < 1.

Hence, using again that γn is strictly decreasing,

λn( f ) < λk( f ) + ε, which implies that

limsupλn( f ) ≤ λk( f ) + ε, n→∞ for every k ≥ 1 and all small enough ε > 0. Therefore,

limsupλn( f ) ≤ inf λk( f ) ≤ liminfλn( f ). n→∞ k∈N n→∞ This discussions is summarized in the following proposition

57 Proposition II.10. It holds,

d¯s(Λ) = lim λn( f ) = inf λn( f ). n→∞ n∈N

The real number d¯s(Λ) = d¯s( f ;Λ) is called upper stable dimension of the −1 horseshoes Λ associated to f . d¯u(Λ) = d¯u( f ;Λ) := d¯s( f ,Λ) is called upper unstable dimension of the horseshoe Λ associated to f .

Remark 10. By symmetry many of the properties satisfied by the upper stable dimension is also satisfied by the upper unstable dimension. Then, it is not restriction to fix our attention only in the upper stable dimension and throughout this work we will do so.

Remark 11. The upper stable dimension does not depends on the choice of the Markov partition P. Take x ∈ Λ and note that

∩W s (x) ⊆ [ V (b) ∩W s (x) Λ ε0 f ε0 b∈W−; |b|=n for every n ≥ 1. Moreover, for b ∈ W−, |b| = n

V (b) ∩W s (x) ≤ Dˆ (b). diam f ε0 s Using the Proposition II.8 we have that

( f ) diamW s (x) ∩V (b)λn ≤ Dˆ (b)λn( f ) < ∞ (7.2) ∑ ε0 f ∑ s b∈W−; |b|=n b∈W−; |b|=n Using the deﬁnition of Hausdorff dimension and that fact that the set

{W s (x) ∩V (b) b ∈ W−,|b| = n} ε0 f ; ∩W s (x) n ∈ is a cover of Λ ε0 , for every N, 7.2 gives us the following proposition. Proposition II.11. For every x ∈ Λ

∩W s (x) ≤ d¯ ( ) dimH Λ ε0 s Λ (7.3) The left hand side of 7.3 is usually called stable dimension of the horseshoe Λ at x. At some circumstances these numbers are constant with respect to x (for instance for surface C2 horseshoes, see ([39], page 60) and in these cases the d ( ) = ∩W s (x) stable dimension number s Λ dimH Λ ε0 is called of the horseshoe Λ. Using unstable instead of stable manifolds it is possible to deﬁne unstable dimensions.

58 Remark 12. The Proposition 7.3 above is the motivation to the word ”upper” in the deﬁnition of the ”upper stable dimension”, d¯s(Λ).

Remark 13. It is not known when the numbers ∩W s (x) are indepen- dimH Λ ε0 dent of x ∈ Λ, for general horseshoes (inside manifolds M with dim(M) > 2).

1 Consider U ( f ) ⊆ Diff (M) a small neighborhood of f and let Λg be the hyperbolic continuation of Λ for g ∈ U ( f ).

Proposition II.12. The map

g ∈ U ( f ) 7−→ d¯s(Λg) is upper semicontinuous.

Proof. Observe that the function g ∈ U ( f ) 7−→ λn(g) is, by deﬁnition, continuous for every n ≥ 1. Since the inﬁmum of continuous functions is an upper semicontinuous function we have the result.

59 8 HORSESHOES WITH S-SPLITTING AND ONE DIMENSION WEAK STABLE BUNDLE

Let Λ ⊆ M be a horseshoe for diffeomorphism f ∈ Diffr(M), r > 1. In this ss w u section we will assume that Λ has a s-splitting TΛM = E ⊕ E ⊕ E and that dim(Ew) = 1. Let U ( f ) ⊂ Diffr(M) be a small neighborhood of f such that Λ has a hyperbolic continuation (with s-splitting and one dimension weak stable direction), Λg, for every g ∈ U ( f ). We will also consider P a Markov partition for Λ with alphabet A. As it was seen at Section 7 this implies that the homeomorphisms ig : Σ → Λg are (bi)-Holder¨ continuous (with uniformly Holder¨ constant and exponents) and conjugates (σ,Λ) and (g|Λg ,Λg) for every g ∈ U ( f ). Note that, for any g ∈ U ( f ) and for every x ∈ Λg

dg(x)| s = dg(x)| w . Eg(x) Eg (x)

This is due to the fact that Λg has a s-splitting. Another simple observation the we will use repeatedly in this work without make any mention and comes from the fact that the weak stable bundle is one dimension is that n−1 n j dg (x)| w = dg(g (x)| w j . Eg (x) ∏ E (g (x))) j=0 This is one of the key properties of these types of horseshoes and allow us to change the problem from a high dimension to a one dimensional environment where conformality holds. Since the all the previously mentioned bundles, in particular the weak stable w bundle, Eg , are also Holder¨ continuous, with uniform Holder¨ constant and exponent for every g ∈ U ( f ), we have that there exist Cw > 0 and αw ∈ (0,1) such that

αw log dg(y)| w − log dg(z)| w ≤ C d(y,z) (8.1) Eg (y) Eg (z) w for every y,z ∈ Λg and for every g ∈ U ( f ). This information will be important to us in the next section.

60 8.1 Bounded distortion estimates

Some of the results in this section already appeared in [28]. For completeness we will repeat here these results (with a bit more amount of details) with the objective of keep consistent the statement of the results and the notation already introduced in the previous sections. Remember that in Proposition II.7 we proved that for every b,c ∈ W−

Ds(bc) ≤ Ds(b) · Ds(c).

The intention now is to give a lower bound for Ds(bc) in terms of the product Ds(b) · Ds(c). The ﬁrst step for this is following proposition: Proposition II.13. It holds

u (i) There exist C1 > 0 and α ∈ (0,1) such that for every y,z ∈ Λg, y ∈ Wg (z) and every g ∈ U ( f ),

n −n+1 dg (g (y))|Ew(g−n+1(y)) α g α exp(−C1d(y,z) ) ≤ ≤ exp(C1d(y,z) ) n −n+1 dg (g (z))| w −n+1 Eg (g (z)) for every n ≥ 1.

0 0 0 s 0 (ii) There exists C2 > 0 such that for every y ,z ∈ Λg, y ∈ W (z ), and every g ∈ U ( f ), n 0 dg (y )| w 0 Eg (y ) exp(−C2) ≤ ≤ exp(C2) n 0 dg (z )| w 0 Eg (z ) for every n ≥ 1.

0 0 Proof. For every n ≥ 1 and y ,z ∈ Λg, using 8.1, we have

 n 0  dg (y )| w 0 Eg (y ) log  = n 0 dg (z )| w 0 Eg (z ) n−1 ! n−1 ! j 0 j 0 = log dg(g (y ))| w j 0 − log dg(g (z ))| w j 0 ∏ Eg (g (y )) ∏ Eg (g (z )) j=0 j=0 (8.2) n−1 j 0 j 0 ≤ log dg(g (y ))| w j 0 − log dg(g (z ))| w j 0 ∑ Eg (g (y )) Eg (g (z )) j=0 n−1 j 0 j 0 αw ≤ Cw ∑ d(g (y ),g (z )) . j=0

61 0 −n+1 −n+1 s Write y = g (y) and z = g (z), with y ∈ Wg (z), and observe that

n−1 n−1 ∑ d(g j(y0),g j(z0))αw = ∑ d(g−n+1+ j(y),g−n+1+ j(z))αw j=0 j=0 n−1 = ∑ d(g− j(y),g− j(z))αw j=0 n−1 αw − j αw ≤ dg (ξ j)| u · d(y,z) ∑ Eg (ξ j) (8.3) j=0 n−1 − jαw α ≤ d(y,z) w · m(dg| u ) ∑ inf Eg j=0 Λg ∞ − jαw ! α ≤ m(dg| u ) · d(y,z) w ∑ inf Eg j=0 Λg

Consider C1 > 0 such that ∞ − jα C ≥ C · ( m(dg| u )) w 1 w ∑ inf Eg j=0 Λg for every g ∈ U ( f ). So, by 8.2 and 8.3, we have that for every y,z ∈ Λg with y ∈ W u(z),

 n 0  dg (y )| w 0 Eg (y ) αw log  ≤ C1 · d(y,z) n 0 dg (z )| w 0 Eg (z ) This proves the item (i). To conclude item (ii) assume that y0 ∈ W s(z0), and observe that in this case there exists C > 0 and λ ∈ (0,1) such that

d(g j(y0),g j(z0)) ≤ Cλ j for every g ∈ U ( f ). Then, again by 8.2 we have

 n 0  dg (y )| w 0 n−1 ∞ Eg (y ) jαw jαw log  ≤ CwC λ ≤ CwC λ n 0 ∑ ∑ dg (z )| w 0 j=0 j=0 Eg (z )

∞ jαw Setting C2 = CwC ∑ j=0 λ we conclude the proposition. Proposition II.14. There exits C˜ > 1 such that

− −|b| (i) For every b ∈ W and every y ∈ g (Vg(b)) we have

|b| D (g;b) ≤ C˜ dg (y)| w , s Eg (y) for every g ∈ U ( f ).

62 − − (ii) For every n ≥ 1 and any b = b−n+1 ...b−1b0 ∈ W , with b−n+1,...,b0 ∈ W (possibly with different sizes), we have

D (g;b) C˜−n ≤ s . ≤ 1 n−1 ∏ Ds(g;b−i) i=0

−|b| −|b| Proof. Fix y ∈ g (Vg(b)). Then, for every z ∈ g (Vg(b)), we have

|b| |a| dg (z)| w ≤ exp(C d(y,z)) dg (y)| w Eg (z) 1 Eg (y) |a| ≤ C˜ dg (y)| w Eg (y) which implies that

|b| D (g;b) ≤ C˜ dg (y)| w , s Eg (y)

−|b| for every y ∈ g (Vg(b)). This concludes (i). − −|b| Now take, as in (ii), b = b−n+1 ...b0 ∈ W . Consider y ∈ g (Vg(b)) and −|b | note that this implies that y ∈ g −n+1 (Vg(b−n+1)). More generally, for every 1 ≤ i ≤ n − 1 we have

|b−n+1...b−n+i| −|b−n+i+1| g (y) ∈ g (Vg(b−n+i+1)).

So, using (i), we have

Therefore, D (g;b) s ≥ C˜−n. n−1 ∏ Ds(g;b−i) i=0 The other inequality of (ii) is proved directly by II.7.

63 8.2 Applications of bounded distortion

The ﬁrst application that we give is the relation between the stable diameter of an word b ∈ W− and its size |b|. Remember the constants ν and τˆ introduced in 7.1. Proposition II.15. For every word b ∈ W− we have

|b| |b| τ ≤ Ds(g;b) ≤ τˆ , for every g ∈ U ( f ).

− Proof. Write b = b−|b|+1 ...b0 ∈ W . Using the estimates in the Proposition II.14, item (ii), we have |b|−1 |b| |b| D (g;b) ≤ D (g;b ) ≤ sup dg| w ≤ τˆ , (8.4) s ∏ s −i Eg i=0 Λ −|b| and for any y ∈ g (Vg(b)) |b| |b| |b| Ds(g;b) ≥ dg (y)|Ew(y) ≥ inf dg|Ew ≥ τ . g Λ g

Another application is regarding the relation between Ds(b) and Dˆ s(b). Re- member that in II.8 we proved that

Dˆ s(b) ≤ dim(Λ) · Ds(b) for every b ∈ W−. Now with the hypothesis of one dimensional weak stable bundle we are able to guarantee that Dˆ s(b) and Ds(b) are indeed equivalent. Proposition II.16. There exists a constant C > 0 such that

Ds(b) ≤ C · Dˆ s(b) for every b ∈ W− with |b| large enough.

s Proof. Take y ∈ Vf (b) and x,z ∈ Wε (y). For |b| large enough we can ﬁnd x,z such that d( f −|b|(x), f −|b|(z)) ≥ 1 and that there exists a path tangent to the weak stable bundle connecting x,z (remember that Ew is one dimensional). Applying the mean value theorem to the composition of f −|b| with this path we have that

−|b| −|b| −|b| d( f (x), f (z)) ≤ d f (x )| w d(x,z) 0 E (x0) −|b| s ≤ d f (x )| w diam W (y) ∩V (b) . 0 E (x0) ε f

64 This implies that

|b| |b| d f (x )| w = m( f (x )| w ) ≤ Dˆ (b) 0 E (x0) 0 E (x0) s Using the item (ii) of the Proposition II.13 (in fact we are using that the Proposi- tion still holds for a small neighborhood of the hyperbolic sets Λg) we conclude that

Ds(b) ≤ C · Dˆ s(b). This ﬁnishes this proposition.

We ﬁnish this section giving a proof of the continuity of the upper stable dimension in the context of horseshoes with one dimensional weak stable bundle.

Theorem II.17 ([28]). If Λ is a horseshoe with s-splitting and one dimensional weak stable bundle for f ∈ Diffr(M), r > 1, then the map

1+ g ∈ U ( f ) ⊂ Diff (M) 7−→ d¯s(Λg) is continuous.

Proof. It is enough, by II.12, to prove that this function is lower semicontinuous.

Let (a,c) ∈ A be an admissible pair and consider the numbers µn(g) deﬁned by

µn(g) ∑ Ds(g;b) = 1 b∈W−; |b|=n b0=c, b−n+1=a Note that

µn(g) µn(g) 1 = ∑ Ds(g;b) ≤ ∑ Ds(g;b) b∈W−; |b|=n b∈W−; |b|=n b0=c, b−n+1=a which implies that λn(g) ≥ µn(g). Observe also that the functions g ∈ U ( f ) 7−→ µn(g) are continuous for every n ≥ 1. − p r s p r s Fix m0 ∈ N. Any word b ∈ W can be written as b = b b b with, b ,b ,b ∈ − p s r W , |b | = m0,|b | = m0 and |b | = n − 2m0 (here the superscripts p, r and s stand for prefix, suffix and radical of the word b). Observe that given any word 0 − 0 − 0 r b ∈ W , |b | = n−2m0 we can find b ∈ W with |b| = n such that b = b . Since p r s the horseshoe Λg is mixing we can take m0 ∈ N such that the word b = b b b can be chosen satisfying that br = b0, bp = c and bs = a. −m0 0 Given ε > 0 consider n0 ∈ N such that

τˆ(n0−2m0)ε ≤ C˜−3 · τ2m0 , (8.5)

65 and

τˆn0ε ≤ C˜−1, (8.6) where C˜ is the constant in the Proposition II.14 and τ < τˆ are the constants deﬁned in 7.1. Claim 1.

λn0−2m0 (g) ≤ (1 + ε)µn0 (g) ≤ (1 + ε)λn0 (g). Note that we already have the second inequality of the claim. To proof the ﬁrst 0 − 0 − inequality take b ∈ W , |b | = n0 − 2m0, and consider a word b ∈ W such that br = b0. Then, by II.15, 8.5 and by Proposition II.14, item (ii), we have

II.15 0 1+ε (n −2m )ε 0 Ds(g;b ) ≤ τˆ 0 0 · Ds(g;b ) 8.5 −3 2m 0 ≤ C˜ · τ 0 · Ds(g;b ) II.15 −3 p r s ≤ C˜ · Ds(g;b ) · Ds(g;b ) · Ds(g;b ) II.14 ≤ Ds(g;b).

Hence,

0 (1+ε)µn (g) µn (g) ∑ Ds(g;b ) 0 ≤ ∑ Ds(g;b) 0 = 1, 0 − − b ∈W ; b∈W ; |b|=n0, |b0|=n −2m b =c, b =a 0 0 0 −n0+1 which implies that

λn0−2m0 (g) ≤ (1 + ε)µn0 (g).

This proves the 1. Claim 2. For every k ≥ 1 we have

(1 − ε)µn0 (g) ≤ λn0k(g)

66 Indeed, using 8.6, II.15 and again the Proposition II.14, item (ii), we have

 k

 (g)  µn0  1 =  ∑ Ds(g;b)  −  b∈W ; |b|=n0,  b =c, b =a 0 −n0+1 II.14 ˜kµn (g) ˆ µn (g) ≤ C 0 ∑ Ds(g;b) 0 − bˆ∈W ; |bˆ|=n0k, bˆ =c, bˆ =a 0 −n0k+1 8.6 −n0εkµn (g) ˆ µn (g) ≤ ∑ τˆ 0 Ds(g;b) 0 − bˆ∈W ; |bˆ|=n0k, bˆ =c, bˆ =a 0 −n0k+1 II.15 ˆ (1−ε)µn (g) ˆ (1−ε)µn (g) ≤ ∑ Ds(g;b) 0 ≤ ∑ Ds(g;b) 0 − − bˆ∈W ; |bˆ|=n0k, bˆ∈W ; |bˆ|=kn0 bˆ =c, bˆ =a 0 −n0k+1 which implies that

(1 − ε)µn0 (g) ≤ λkn0 (g), for every k ≥ 1. This concludes the Claim 2. Making k → ∞ in the expression of the Claim 2 we have

¯ (1 − ε)µn0 (g) ≤ ds(Λg).

Then, by Claim 1, we have that 1 − ε d¯s(Λg) ≥ λ (g) 1 + ε n0−2m0 which implies that 1 − ε 1 − ε liminfd¯s(Λg) ≥ λn −2m ( f ) ≥ d¯s(Λ f ), g→ f 1 + ε 0 0 1 + ε for every ε > 0. Therefore,

liminfd¯s(Λg) ≥ d¯s(Λ f ). g→ f

67 8.3 Counting pieces

In this section we will use thermodynamics formalism to give estimates of the quantity of elements in sets of pieces of the Markov partition defined by some metric properties which will be important in the subsequent sections. Let X be a compact metric space and consider a continuous map F : X → X. We say that a F-invariant measure µ is a equilibrium state for a continuous map ϕ : X → C, if µ attains the maximum in the variational principle for pressure. In other words, Z P(ϕ) = hµ (F) + ϕ dµ, X where, in the left hand side, appears the pressure of the map ϕ which is defined, in the case that F is expansive (which will be our case), by ( ! ) 1 n−1 P(F) = lim logsup exp ϕ(F j(x)) ; E (n,ε) separable set . n→∞ ∑ ∑ n x∈E j=0 (8.7) This limits does not depend on the choice of the small enough ε > 0. Another important remark is that when ϕ ≡ 0 its pressure coincides with the topological entropy of the system F and the equilibrium state in this case is a measure of maximal entropy. We say that a F-invariant measure µ is a Gibbs state if there exists some positive constant C such that µ (B(x,ε;n)) C−1 ≤ ≤ C, n−1 ! exp ∑ ϕ(F j(x)) + nP(ϕ) j=0 for every x ∈ X, n ∈ N and for every sufficient small ε > 0. Here n−1 \ B(x,ε;n) = F− j(B(F j(x),ε)) j=0 denotes the Bowen’s ball. For more details about pressure and entropy of a system see [49]. In [14], Bowen proved that if F is an expansive homeomorphism satisfying the specification property, then every Holder¨ continuous map ϕ : X → C has a unique equilibrium states µϕ which is a Gibbs state. Specification means that for every δ > 0, there exists p(δ) ∈ N such that if I1,...,In are integer intervals, Ij ⊆ [a,b] and d(Ii,Ij) ≥ p(δ) for every i 6=

68 b−1+p(δ) j and x1,...,xn ∈ X, then there exists x ∈ X such that f (x) = x and k k d( f (x), f (xi)) < δ, for every k ∈ Ii. Consider the data, X = Σ, F = σ −1 and ϕ deﬁned by

ϕ (θ) = t log d f (i (θ))| w t f E (i f (θ))

−1 Note that since f |Λ is topologically mixing the map σ : Σ → Σ has the speciﬁ- cation property. Applying the theorem to these data we have that there exists a unique equilibrium state µϕs measure in Σ which satisﬁes that there exists C > 0 such that

µ (b) C−1 ≤ ϕt ≤ C, (8.8) exp(Snϕt(θ) + nP(ϕt))

+ For every n ≤ 1 and for every word b ∈ W , such that θ ∈ b. Here Snϕs = n−1 j ∑ j=0 ϕs ◦ F . Proposition II.18. P( ) = 0 ϕd¯s(Λ) Proof. First note that

n−1 ! −1 − j−1 S ϕ (σ (θ)) = t log d f ( f (i (θ)))| w − j−1 n t ∏ f E ( f (i f (θ))) j=0 n −n = t log d f ( f (i (θ)))| w −n . f E ( f (i f (θ)))

So, calling i f (θ) = x ∈ Vf (b), we have

−1 n −n s exp Snϕt(σ (θ)) = d f ( f (x))|Ew( f −n(x)) .

Taking ε = 1 in the deﬁnition of the pressure in 8.7 and noting that the supremum of ∑ exp(Snϕt(θ)) over the (n,1)-separable sets E is attained for the set {b ∈ θ∈E W−;|b| = n} (which is (n,1)-separable) gives us that

∗ 1 n −n d¯s(Λ) P(ϕ ¯ ) = lim log d f ( f (xb))|Ew( f −n(x )) . ds(Λ) n→∞ n ∑ b b∈W−;|b|=n

− In ∗ xb is any element in b ∈ W , |b| = n. Note that by Proposition II.14, item (i) we have that

−1 d¯s(Λ) n −n d¯s(Λ) ˜ w C Ds(b) ≤ d f ( f (xb))|E (xb) ≤ Ds(b) ,

69 which implies that

−1 n −n C˜ (d¯ ( )) ≤ d f ( f (x ))| w ≤ (d¯ ( )). γn s Λ ∑ b E (xb) γn s Λ b∈W−; |b|=n So, 1 1 −o (1) + limsup log (d¯ ( )) ≤ P( ) ≤ liminf log (d¯ ( )). n γn s Λ ϕd¯s(Λ) γn s Λ n→∞ n n→∞ n But,

d¯s(Λ)−λn( f ) d¯s(Λ)−λn( f ) τ ≤ γn(d¯s(Λ)) ≤ τˆ

This conclues that P( ) = 0. ϕd¯s(Λ) A directly corollary is expressed in the following proposition.

Proposition II.19. There exists C > 0 such that

µϕ ¯ (b) C−1 ≤ ds(Λ) ≤ C d¯ (Λ) Ds(b) s for every b ∈ W−

Proof. Follows from 8.8.

The next proposition is the goal of this section and will be one of the most important tools in this work. The first item gives an estimation of the amount of boxes which has a diameter of order ρ ∈ (0,1) in terms of ρ and the upper stable dimension. The second item, in the other hand, estimates the amount of boxes which has diameter of order ρ but with fixed distant from a fixed box.

Proposition II.20. Consider positive constants A0 and A1.

(i) There exists C1 > 0 such that for every ρ ∈ (0,1)

− −d¯s(Λ) b ∈ W ; A0ρ ≤ Ds(b) ≤ A1ρ ≤ C1ρ ;

(ii) There exists a positive constant C2 such that for every natural k ≥ 1, every − number ρ ∈ (0,1) and any box c ∈ W satisfying A0ρ ≤ Ds(c) ≤ A1ρ we have n o − −k−1 −k b ∈ W ;A0ρ ≤ Ds(b) ≤ A1ρ,2 ≤ Ds(b ∧ c) < 2 −d¯ (Λ) −1 −1 k s ≤ C2 log(A0 ρ ) 2 ρ .

70 Proof. Set − S(ρ;A0,A1) = b ∈ W ; A0ρ ≤ Ds(b) ≤ A1ρ . Then,   [ 1 ≥ µ b = µ (b) ϕd¯s(Λ)   ∑ ϕd¯s(Λ) b∈S(ρ;A0,A1) b∈S(ρ;A0,A1) II.19 −1 d¯s(Λ) ≥ C Ds(b) |S(ρ;A0,A1)|

−1 d¯s(Λ) ≥ C (A0ρ) |S(ρ;A0,A1)| which implies that ¯ −ds(Λ) −d¯s(Λ) |S(ρ;A0,A1)| ≤ CA0 · ρ . This proves the item (i). − −k−1 −k To prove the item (ii) ﬁx d ∈ W such that 2 ≤ Ds(d) < 2 . Deﬁne the set − Td(ρ) = bˆ ∈ W ; A0ρ ≤ Ds(bˆ · d) ≤ A1ρ . Claim 3. Take C˜ as in Proposition II.14. Then

k −2 Td(ρ) ⊆ S(2 ρ;A0,2C˜ A1)

Indeed, using the Proposition II.14, it is simple to see that if bˆ ∈ Td(ρ) ˆ −2 Ds(b · d) −2 k+1 Ds(bˆ) ≤ C˜ ≤ C˜ A1ρ2 , Ds(d) and ˆ Ds(b · d) k Ds(bˆ) ≥ ≥ A02 ρ. Ds(d) This ﬁnishes the Claim. Now observe that the claim implies that ¯ −ds(Λ) −k −d¯s(Λ) |Td(ρ)| ≤ CA0 · (2 ρ) . (8.9) Take c as in the statement of the item (ii). Then,

|c| A0ρ ≤ Ds(c) ≤ τˆ which implies that log(A ρ) |c| ≤ 0 logτˆ

71 To ﬁnish the proof it enough to observe that the set

n − −k−1 −ko b ∈ W ;A0ρ ≤ Ds(b) ≤ A1ρ,2 ≤ Ds(b ∧ c) < 2

− is contained in the union of sets Td(ρ) over d ∈ W where c ⊂ d. So, using 8.9, we have n o − −k−1 −k b ∈ W ;A0ρ ≤ Ds(b) ≤ A1ρ,2 ≤ Ds(b ∧ c) < 2 ¯ −ds(Λ) −d¯s(Λ) ≤ |c|CA0 · ρ d¯s(Λ) CA ¯ ≤ 0 · log(A−1ρ−1)(2kρ)−ds(Λ). log(τˆ−1) 0

8.4 Extractions

In this section we prove two propositions which will be used to conclude the proof of the Theorem A.

Proposition II.21. Let Λ ⊆ M be a horseshoe for g ∈ Diffr(M), r ≥ 1, and let Γ ( Λ be a closed g-invariant set. Then, given ε > 0, there exists a subhorseshoe 0 ¯ 0 ¯ 0 Λ ⊆ Λ such that ds(Λ ) ≥ ds(Λ) − ε and Λ ∩ Γ = ∅.

This proposition is an adaptation of the Proposition in the Section 4.2.2 in [30]. We will leave the proof of this proposition to the end of this section. First we will state and proof (using Proposition II.21) a proposition which suits better our intentions in the proof of the Main theorem:

Proposition II.22. Consider f ∈ Diffr(M) and let Λ be a horseshoe for f k, for some k ≥ 2, such that f −i(Λ) 6= Λ, for every 1 ≤ i < k. Given ε > 0 there exist 0 k0 ≤ k and a horseshoe Λ0 ⊆ Λ, for f k , such that

0 1. d¯s(Λ ) ≥ d¯s(Λ) − ε;

−i 0 − j 0 0 2.f (Λ ) ∩ f (Λ ) = ∅ for every i 6= j, 1 ≤ i, j ≤ k − 1.

− j 0 Proof. Set Λ0 = Λ, j1 = 1, Γ0 = f 1 (Λ0) ∩ Λ0 and ε = ε/k. Observe that Γ0 ( k k Λ0 is closed and f -invariant. Then, applying the Proposition II.21 with g = f , k we have that there exists a subhorseshoe Λ1 of Λ0 for f such that d¯s(Λ1) ≥ ¯ 0 − j1 ds(Λ0) − ε and Λ1 ∩ Γ0 = ∅. This implies that f (Λ1) ∩ Λ1 = ∅.

72 Consider the number

−i j2 = max m ≥ 1; f (Λ1) ∩ Λ1 = ∅ for every 1 ≤ i < m .

− j 0 Note that j2 > j1. If f 2 (Λ1) = Λ1, we take k = j2 and the result is proved. In − j k the other case we can deﬁne Γ1 = f 2 (Λ1) ∩ Λ1 ( Λ1 which is compact and f - invariant. Then, by Proposition II.21, we have that there exists Λ2 subhorseshoe of ¯ ¯ 0 −i Λ1 such that ds(Λ2) ≥ ds(Λ1) − ε and Λ1 ∩ Γ1 = ∅ which implies that f (Λ2) ∩ Λ2 = ∅ for every 1 ≤ i ≤ j2. Suppose that we ﬁnd 1 = j1 < j2 < ··· < jl and a nested sequence of sub- k −i horseshoes of Λ0, Λ0 ) Λ1 ) ··· ) Λl, for f , such that f (Λl) ∩ Λl = ∅, for 0 every 1 ≤ i ≤ jl and d¯s(Λl) ≥ d¯s(Λl−1) − ε . Consider

−i jl+1 = max m ≥ 1; f (Λl) ∩ Λl = ∅, for every 1 ≤ i < m .

− j Note that jl+1 > jl. If f l+1 (Λl) = Λl there is nothing more to do. It enough to 0 − j − j take k = jl+1. Assume then that f l+1 (Λl) 6= Λl and set Γl = f l+1 (Λl)∩Λl ( Λl which is closed and f k-invariant. Then using again the Proposition II.21 we have k 0 that there exists a subhorseshoe Λl+1 of Λl for f such that d¯s(Λl+1) ≥ d¯s(Λl)−ε −i and Λl+1 ∩ Γl = ∅ which implies that f (Λl+1) ∩ Λl+1 = ∅, for every 1 ≤ i ≤ jl+1. Continuing this inductive process, noting that ji > i, either stopping the pro- 0 cess before of k or exhausting the set {1,...,k}, we ﬁnd k equals to some jm and 0 k a subhorseshoe Λ = Λ jm of Λ0 for f such that

0 0 d¯s(Λ ) ≥ d¯s(Λ0) − kε = d¯s(Λ) − ε,

−i 0 0 0 f (Λ ) ∩ Λ = ∅, for every 1 ≤ i ≤ k − 1 which conclude the result.

Proof. Proposition II.21: Consider a word a ∈ W such that Γ∩a = ∅. Set |a| = M and M˜ much large than M to be chosen later. Consider the set Σ˜ of sequences θ ∈ Σ such that any sub-word of θ of size M˜ contains a. In other words,   \ [ − j−k Σ˜ =  σ (a). k∈Z | j|≤M˜

Set Λ˜ g = ig(Σ˜) ⊂ Λg where ig : Σ → Λg is the conjugations between the symbolic dynamics (σ,Σ) and (g,Λg). Observe that Λ˜ g is a subhorseshoe of Λg for g. Follows direct by deﬁnition that Λ˜ is transitive and locally maximal, moreover since Γ is f -invariant and Γ ∩ a = ∅, we have Γ ∩ Σ˜ = ∅.

73 Let W˜ and W˜ − be, respectively, the set of ﬁnite words and the sets of negative words which are allowed in the set Σ˜. Consider M¯ > M˜ and ﬁx J1,...,Jk ⊂ [−M¯ + 1,0] ∩ Z, k = 2M¯ /M˜ disjoints intervals with length l = M selected such that d(Ji,Ji+1) ≤ M˜ (it is enough to subdivide [−M¯ + 1,0] in sub intervals of length M˜ /2 (there are 2M¯ /M of them) and pick the intervals Ji spread in this intervals). − We will write any word b ∈ W , |b| = M¯ , as b = c−kd−k ...c−1d−1, where − b−i, d−i ∈ W and |c−i| = M for every 1 ≤ i ≤ k, moreover, if we write b = b−|b|+1 ...b0 and Ji = {−li −M +1,...,−li}, then b−li−M+1 ...b−li = c−i, for every 1 ≤ i ≤ k. − We will also use the following notation: for any word b = b−M¯ +1 ...b0 ∈ W and Ji = {−li − M + 1,...,−li} write

b = c−kd−k ...c−1d−1,

− where b−i,d−i ∈ W , |c−i| = M and b−li−M+1 ...b−li = c−i, for every 1 ≤ i ≤ k. Consider the map ν : W− → W− deﬁne by

ν(b) = ad−k ...ad−1,

− − − where b = c−kd−k ...c−1d−1. Set V˜ = ν(W ) and observe that V˜ is a subset of W˜ − which satisﬁes that for every b˜ ∈ V˜ −

−1 kM ν (b˜) ≤ |A| . (8.10)

− Note also that for any b = c−kd−k ...c−1d−1 ∈ W , by II.14 and II.15, we have

II.14 k k Ds(g;b) ≤ ∏Ds(g;d−i) · ∏Ds(g;c−i) i=1 i=1 II.14 k ˜2k −k ≤ C · Ds(g;a) · Ds(g;ν(b)) · ∏Ds(g;c−i) (8.11) i=1 II.15 kM τˆ 2k ≤ ·C˜ · Ds(g;ν(b)) τ

Consider the numbers λ˜ n(g) which converges to d¯s(Λ˜ g) and satisﬁes

˜ ˜ λn(g) ∑ Ds(g;b) = 1. b˜∈W˜ −;|b˜|=M¯

74 So, using 8.10 and 8.11, we have

λ ¯ (g) λ ¯ (g) 1 = ∑ Ds(g;b) M = ∑ ∑ Ds(g;b) M b∈W−;|b|=M¯ b˜∈V˜ −;|b˜|=M¯ b∈ν−1(b˜) 8.11 ˆ kMλM¯ τ 2kλ ¯ −1 λ˜ ¯ (g) λ ¯ (g)−λ˜ ¯ (g) ≤ ·C˜ M · |ν (b˜)| Ds(g;b˜) M · Ds(g;b˜) M M τ ∑ b˜∈V˜ −;|b˜|=M¯

kMλM¯ 8.10 τˆ ¯ ˜ ≤ ·C˜2kλM¯ · |A|kM · τˆM(λM¯ (g)−λM¯ (g)). τ

This implies that

τˆ 0 ≤ kMλ ¯ (g)log + 2kλ ¯ (g)log(C˜) + kM log(A) + M¯ (λ ¯ (g) − λ˜ ¯ (g))log(τˆ) M τ M M M

Dividing by M¯ and remembering that k = 2M¯ /M˜ we have

2M τˆ 4 2M 0 ≤ λ ¯ (g)log + λ ¯ (g)log(C˜) + log(A) + (λ ¯ (g) − λ˜ ¯ (g))log(τˆ) M˜ M τ M˜ M M˜ M M

Sending M¯ to inﬁnity and dividing by log(τˆ) (remember that τˆ < 1) we have

2M τˆ 4 2M 0 ≥ d¯s(Λg)log + d¯s(Λg)log(C˜) + log(A) + d¯s(Λg) − d¯s(Λ˜ g) M˜ log(τˆ) τ M˜ log(τˆ) M˜ log(τˆ)

So, for every ε > 0, we can ﬁnd M˜ = M˜ (M,ε) such that

d¯s(Λ˜ g) ≥ d¯s(Λ) − ε.

75 9 RANDOM PERTURBATIONS AND PROJECTIONS

The goal of this section is to perform perturbations (with higher regularities) satisfying some fixed property, say ”P”. The idea is to use the totally disconnected character of horseshoes to address the problem randomly and prove that with positive probability the property P is true. In particular, there will exist, at least one diffeomorphisms satisfying the property P. The advantage of use this technique is that now we are allowed to use, although basic, extremely powerful tools of the probability theory to approach the problem. The idea of use probabilistic tools to guarantee existence of an object, as described above, is not new. This appeared, at the beginning, in the works of Erdos¨ (see for example [19]) to attack, easily, problems in graph theory and, since then, has been used a lot in many areas of mathematics (mainly in combinatorics) and is called the Probabilistic method. We want to stress here that, despite the simplicity of the argument, this is a powerful technique with a lot of skilful applications. We start with some definitions that will be used throughout this section (and also in the subsequent sections) and define the general setting of the random perturbations. Let Λ ⊂ M be a horseshoes for a diffeomorphism f ∈ Diffr(M), r > 1. Assume ss w u w that Λ has a s-splitting TΛM = E ⊕ E ⊕ E and that dim(E ) = 1. Let P be a Markov partition for Λ with alphabet A. We will assume, for continuity purposes, that the diameter of the elements of P is as small as we need. We use here the notation introduced in the previous sections.

For each a ∈ A consider a f -periodic point pa ∈ P(a) and a smooth submanifold Ha of M which satisﬁes:

u • dim(Ha) = dim(E ) + 1;

w u • Ha is tangent to E (pa) ⊕ E (pa);

ss • Ha is transverse to the strong stable lamination W in P(a)

ss • For any x ∈ P(a), |Wloc(x) ∩ Ha| = 1.

76 Note that the above properties remains true for any g in a small open neighborhood U ( f ) ⊂ Diff1(M) of f . Set [ H = Ha a∈A w w We will fix, for future purposes, for each a ∈ Q, a unit vector ea ∈ E (pa). For each a ∈ A and g ∈ U ( f ), define πg,a : Pg(a) → Ha the projection along the strong stable leaf, in a over Ha. When we are working with a fixed stable s + + + leaf Wg,loc(θ ), for some θ ∈ Σ , θ0 = a, we use the notation πg,θ + for the s + s + restriction of πg,a to Wg,loc(θ ). Observe that the image of Wg,loc(θ ) by πg,θ + is s + a one dimensional manifold given by the intersection of Wg,loc(θ ) with Ha.

9.1 Random perturbations

Here we give the general framework of the random perturbations. For each word a ∈ W and δ > 0 set

U(a;δ) = {x ∈ M; d(x;P(a)) ≤ δ}, the δ-neighborhood of the set P(a). Let Q ⊂ W be a ﬁnite partition of Σ. Note that Q induces, naturally, a ﬁnite partition of Λ by boxes P(a), a ∈ Q. By compactness of the boxes P(a), a ∈ Q, we have that there exists ξ > 0 such that d(P(a),P(b)) ≥ ξ, for pair a,b ∈ Q, a 6= b.

Take a random vector X = (Xa)a∈Q with i.i.d. coordinates and, for each 1 ≤ ∞ s ≤ r, let ψX : M → M be a C diffeomorphism satisfying the following properties:

s • ψX is close to the identity id : M → M in the C topology;

• Outside of ∪a∈QU(a;ξ/2), ψX coincides with the identity id map;

• For each x ∈ U(a;ξ/2) the function

|Q|−1 v ∈ R 7−→ ψ(Xa,v)(x)

is constant.

The last property above basically say to us that we can interpret φX |U(a;ξ/2) as a random function which only depends on the random variable Xa. The diffeomorphism

r fX = f ◦ ψX ∈ Diff (M)

77 is called a random perturbation of f in the Cs topology.

Observe that the diffeomorphism ψX is itself a random perturbations of the identity, id, in the Cs topology. In the next section we will introduce a family of random perturbations of f , in the Cr topology, which is suitable for our purposes in this work.

9.2 Translating randomly in the weak stable directions

Let ρ be a positive real number. This will be our main parameter in our construction and we will always think in ρ close to 0. Fix positive constants A0 and A1 properly chosen based in the bounded distortion estimates (see 8.1) Consider two positive real numbers ζ and α (you can think as functions of ρ and let Q = Q(ζ,α) ⊂ W be a ﬁnite partition of Σ such that every a = a− ∩a+ ∈ Q satisﬁes that

− |a+| + A0ζ ≤ Ds(a ) ≤ A1ζ and A0α ≤ Ds(σ (a )) ≤ A1α.

For simplicity, we will not carry the constants A0 and A1 and allow ourselves to write only − |a+| + Ds(a ) ζ and Ds(σ (a )) α. In other words, the boxes deﬁned by elements a ∈ Q has stable diameter of order + ζ for a− and order α for σ |a |(a+). −1 Consider the δ0 given by the Proposition II.6, τˆ given in 7.1 and take [i0 ] the −1 Holder¨ exponents of the conjugation, ig , between Λg and Σ for any g ∈ U ( f ) ⊂ Diff1(M).

−ˆ 1 δ0[i0]log(τ ) Proposition II.23. If ζ & ρ 2r , then

δ /2r diam(P(a)) & ρ 0 , for every a ∈ Q.

Proof. Take a = a− ∩ a+ ∈ Q. Using II.15 we have that

1/[i−1] −|a−|/[i−1] diam(P(a)) & diam(a) 0 & 2 0 1 log(τˆ)[i−1] δ /2r & ζ 0 & ρ 0 .

78 Assume for now on that −ˆ 1 δ0[i0]log(τ ) ζ = ρ 2r . (9.1) Using the Preposition II.23 and Proposition II.6 we see that for any pair of words a,b ∈ Q, 1/2r dist(P(a),P(b)) & ρ .

For each a ∈ Q, a ⊂ a0 ∈ A, define a smooth vector field Za of M with the following properties: w 1 Z (x) = e x ∈ U(a 2r / ) • a ρ a0 , for all ;ρ 4 ; 1/2r • Za(x) ≡ 0 outside of U(a;ρ /2); N Set I = [−1/2, 1/2] and N = |Q|. For any t = (ta)a∈Q ∈ I define

Zt = ∑ taZa. a∈Q 1 ∞ Let ϕt ∈ Diff (M) be the time-one map of the ﬂow associated to the ﬁeld Zt. The Cr diffeomorphisms 1 N ft = f ◦ ϕt , t ∈ I are random perturbations for f in the Cr topology such that

1/2 dCr ( f , ft) . ρ , for all t ∈ IN.

Figure 9.1: Random perturbations

When we are dealing with such family of perturbations ( ft)t (and that is what will do from now on) we slightly change our notations writing for exam- ∗ ∗ ple P (a),W (x), Λ , i , φ , π + and π + instead of P (a), W (x), Λ , i , t t,loc t t t t,θ t,θ ft ft ,loc ft ft + + φ ft , π ft ,θ and π ft ,θ . Now we state some properties of the hyperbolic continuation of the points in Λ.

79 Proposition II.24. Take x ∈ Λ and let xt ∈ Λt be the hyperbolic continuation of x. Then,

1. For all n ∈ N, a ∈ Q and s ∈ I. d −n ft (xt) . ρ. dta ta=s

− j 2. Fix a ∈ Q, a ⊂ a0 ∈ A. If f (x) ∈ P(a) and for all 0 ≤ i ≤ n, i 6= j, f −i(x) ∈/ P(a), then

d d x = d f n( f −n(x )) · f −n(x ) + d f j( f − j(x )) · ew . t t t t t t ρ t t t a0 dta dta

Proof. By 6.1 we have that

−n −n −n −n 0 d( ft0 (xt0 ), ft (xt)) = d(φt0 ( f (x)),φt( f (x))) . ρ|t −t |,

N for all n ∈ N and t ∈ I . This implies that d −n ft (xt) . ρ, dta ta=s

N for every n ∈ N, t ∈ I and s ∈ I which concludes the ﬁrst item. To prove the second item, we assume, without loss of generality, that s = 0 and −n i the parameter t = ta ∈ I. Set yn(t) = ft (xt). Since, for 0 ≤ i ≤ n, f (yn(t)) ∈ P(a) if and only if i = n − j, we have,

x = f n(y (t)) = f j−1 ◦ f ( f n− j(y (t))) = f j( f n− j(y (t)) + tew ) = t t n t t t n t n ρ a0 = f n(y (t)) + td f j( f n− j(y (t))) · ew + o (|t|). n ρ n a0 t

Then,

n n xt − x f ◦ yn(t) − f ◦ yn(0) j n− j w ot(|t|) = + ρd f ( f (yn(t))) · e + . t t a0 t Making the limit when t goes to 0 we have

d n −n d −n j − j w xt = d f ( f (x)) · ft (xt) + ρd f ( f (x)) · ea . dt dt t=0 0

80 9.3 Nonrecurrence and transversality condition

Take a = a− ∩ a+ ∈ Q and let β = β(ρ) be a positive real number. We say that a unstable cylinder b ∈ W− is (β,ζ)-nonrecurrent in a if

1. b0 = a0;

2. Ds(b) β;

− 3. For every c ∈ W , Ds(c) ζ we have n o + j + − −|b| + |c| ≤ j ≤ |a |; σ (b ∩ a ) ∩ c 6= ∅ ≤ 1.

− Set Wa (β,ζ) as the set of (β,ζ)-nonrecurrent words in a. For eacha ˆ = aˆ− ∩ aˆ+ ∈ Σ set θˆ + = aˆ+ ∈ Σ+ which is the inﬁnite sequence obtained be right concatenation ofa ˆ+ bya ˆ+ inﬁnitely, i.e.,

θˆ + = aˆ+ · aˆ+ · aˆ+ ....

Proposition II.25. Take a = a− ∩ a+ ∈ Q such that

j σ (θ) ∩ a = ∅ s ˆ + s ˆ + N for every j ≥ 0. Then, Wt,loc(θ ) = W0,loc(θ ) for every t ∈ I . Moreover, the ss s ˆ + strong stable foliation Wt inside W0,loc(θ ) also does not change when we move t ∈ IN.

s ˆ + Remark 14. Although the strong stable foliation inside W0,loc(θ ) does not change s ˆ + when we move t, the points x ∈ W0,loc(θ ) moves. So, in particular, the leaves move when we move t however the whole foliation does not change.

s ˆ + Remark 15. The fact that the strong stable foliation, W0,loc(θ ), is constant when N t moves, implies that the projections πt,θˆ + is equals to π0,θˆ for every t ∈ I .

Proof of Proposition II.25. Assume, without loss of generality, that t = ta ∈ I. Let j + (a j) ⊂ Q be a sequence of boxes such that σ (θˆ ) ⊂ a for every j ≥ 0. Following the same spirit as in 6.6 we have

s ˆ + \ − j \ − j s ˆ + Wt,loc(θ ) = ft (U(a j)) = f (U(a j)) = W0,loc(θ ). j≥0 j≥0

ss s ˆ + We know that by strong stable bundle Et inside of W0,loc(θ ), which is a s ˆ + s ˆ + priori deﬁned only in Λt ∩W0,loc(θ ), is deﬁned the whole leaf W0,loc(θ ) (see

81 ss ss s ˆ + [44] and [45]). Our goal is to prove that Et (x) = E0 (x) for every x ∈ W0,loc(θ ). Indeed, this if the bundles are the same, by unicity of the strong stable foliation, ss s ˆ + ss s ˆ + W ∩W0,loc(θ ) and Wt ∩W0,loc(θ ) must coincide. s ˆ + ss Take x ∈ W0,loc(θ ) and assume that there exists vt ∈ Et (x) which is not in ss ss ss w w ss w E0 (x). Then, there exist v ∈ E0 (x) and v ∈ E0 (x)\{0} such that v = v + v . By the domination between the bundles Ess and Ew (in an open neighborhood of Λ) and the fact that vw 6= 0, we have that

n d f (x) · vt w n ] n ,E ( f (x)) → 0, kd f (x) · vtk when n → ∞. In the other hand if we assume the ρ small enough there exists a positive η > 0 such that

w w ](Et (z),E (z)) < η/2 and n d ft (z) · vt w n ] n ,Et ( ft (z)) ≥ η. kd ft (z) · vtk

N for every z ∈ Λt and t ∈ I . n n n n N Since d f (x) = d ft (x) and f (x) = ft (x) for every t ∈ I and for every n ≥ 0 ss ss s ˆ + we get a contradiction. Therefore Et (x) = E0 (x) for every x ∈ W0,loc(θ ). Set δ [i ]log(τ−1) κ = 0 0 0 2r a positive constant which depends only on a neighborhood of f . Note,by the equation 9.1, that

log(τ−1) ζ log(τˆ−1) = ρκ0 (9.2)

− ˆ + − + − Proposition II.26. Consider b ∈Waˆ (β,ζ), θ ∈ b∩θ and ﬁx a = a ∩a ∈ W . Then

1. There exist positive constants C0, C¯0 and C¯ such that if for some 1 ≤ j < |b| − |a−|, σ − j(θ) ∈ a, then

d −κ0+1 (a) π + (it(θ)) ≤ C¯ ρDs(b− j ···b ) +C¯ρ β; dta t,θˆ 0 0

d −κ0+1 (b) π + (it(θ)) ≥ C ρDs(b− j ···b ) −C¯ρ β, dta t,θˆ 0 0

82 for every t ∈ IN.

2. If σ − j(θ) ∈/ a for all 0 ≤ j ≤ |b| − |a−|, then

d (i ( )) ≤ C¯ −κ0+1 , πt,θˆ + t θ ρ β (9.3) dta ta=s

for every t ∈ IN.

Proof. Assume, again, without loss of generality, that t = ta ∈ I. Set xt = it(θ) and n = |b| − |a−|. By deﬁnition of W−(β,ζ) and the fact that σ − j(θ) ∈ a we have that σ −i(θ) ∈/ a, for every 0 ≤ i ≤ n, i 6= j. This observation allow us to apply the Proposition II.24, item 2 to guarantee that d π ˆ + (xt) = dt t,θ t=0 n −n d −n j − j w = dπ ˆ + (x0) · d f ( f (xt)) · ft (xt) + ρdπ ˆ + (x0) · d f ( f (x0)) · ea , 0,θ dt t=0 0,θ 0

where, as pointed in the Remark 15, we use that the family of projections πt,θˆ + does not change when move the parameter t (remember that we are assuming t = ta ∈ I). Note now that, using the item 1 of the Proposition II.24 and the Proposition II.15, there exists a positive constant C¯, which only depends on f , such that

n −n d −n dπ ˆ + (x0) · d f ( f (xt)) · ft (xt) 0,θ dt t=0 II.24 n −n d −n ≤ dπ ˆ + (x0) · d f ( f (x0))|Es( f −n(x ))) · ft (xt) 0,θ 0 dt t=0 log(τ−1) II.15 − − ≤ C¯ · τ−|a |ρβ ≤ C¯ · ζ log(τˆ−1) ρβ 9.2 ≤ C¯ · ρ−κ0+1β.

Therefore, there exists C¯0 and C0 such that d ¯ ¯ −κ0+1 π ˆ + (xt) ≤ C0ρDs(b− j ···b0) +Cρ β dt t,θ and d ¯ −κ0+1 π ˆ + (xt) ≥ C0ρDs(b− j ···b0) −Cρ β. dt t,θ

83 This concludes the ﬁrst part. If, however, σ −i(θ) never enters in a, then

d n −n d −n π ˆ + (xt) = dπ ˆ + (x0) · d f ( f (xt)) · ft (xt) , dt t,θ t=0 0,θ dt t=0 and so d ¯ −κ0+1 π ˆ + (xt) ≤ Cρ β. dt t,θ

We finish this section with a proposition which is one of main tools in the Marstrand Like argument (see Section 9.4). ˜ − Proposition II.27 (Transversality condition). Consider b,b ∈ Waˆ (β,ζ). Assume that ¯ 4C −κ Ds(b ∧ b˜) ≥ ρ 0 β. C¯0 Then, for any θ ∈ b ∩ θˆ + and θ˜ ∈ b˜ ∩ θˆ + and for any η > 0, we have n N ˜ o η Leb t ∈ I ;|πt,θˆ + (it(θ)) − πt,θˆ + (it(θ))| . η . . ρDs(b ∧ b˜) Proof. Set k = |b ∧ b˜| and take a ∈ Q such that σ −k(θ) ∈ a. We claim that σ − j(θ˜) ∈/ a for any 0 ≤ j ≤ |b˜|. Indeed, first note that, by definition of k we −k + + −k have σ (θ˜) ⊆ a , but θ˜−k−1 6= θ−k−1 = a−1 which implies that σ (θ˜) ∈/ a (observe that, implicitly, we are assuming |a−| ≥ 2). The fact that σ −k(θ˜)+ ⊆ a+ − j ˜ + + ˜ |a+| + gives that σ (θ) ∩a = ∅, for any 0 ≤ j ≤ |b|, j 6= k since Ds(σ (a )) α ˜ ˜ + ˜ and θ ∈ b ∩ θ with b ∈ Wθ + (α,β). This proves the claim. By the Proposition II.26 we have d ˜ ¯ −κ0+1 π ˆ + (it(θ)) ≥ C0ρDs(b ∧ b) −Cρ β, dt t,θ t=s and d ˜ ¯ κ0+1 π ˆ + (it(θ)) ≤ Cρ β. dt t,θ t=s So, if we define ψ : IN → H , by θaˆ0 ˜ ψ(t) = πt,θˆ + (it(θ)) − πt,θˆ + (it(θ)), and use the hypothesis we have d ˜ ψ(t) & ρDs(b ∧ b). (9.4) dta ta=s

84 N−1 Write t = (ta,t˜) ∈ I × I . Using the change of variable theorem and 9.4, we have

  Z Z Z N   ˜ Leb t ∈ I ;|ψ(t)| ≤ η = dt =  dtadt [|ψ|≤η] IN−1 [|ψ(·,t˜)|≤η] Z Z Cη d −1 = ψ (s,t˜) ds dt˜ −Cη dta IN−1 Z Z Cη 1 = dsdt˜ −Cη d ψ(ψ−1(s,t˜),t˜) IN−1 dta 9.4 −1 ˜ −1 N−1 . 2ηρ Ds(b ∧ b) Leb(I ) η . . ρDs(b ∧ b˜)

9.4 Marstrand like argument

Consider, for each θ +Σ+, the quantity n o N ( ) = (b,b˜) ∈ −( , )2 (i (b ∩ +)) ∩ (i (b˜ ∩ +)) 6= , t,θˆ + β Waˆ α β ;πt,θˆ + t θ πt,θˆ + t θ ∅

− which is the number of pairs of pieces in Waˆ (α,β) whose projection by πt,θˆ + intersects each other.

Figure 9.2: Projection of the boxes

85 Proposition II.28. Z −1 −1 −1+κ0(1−d¯s(Λ)) −d¯s(Λ) Nt,θˆ + (β)dt . log(A0 β )ρ β . IN

Proof. For each (b,b˜) ∈ W− (α,β)2 and θ ∈ b ∩ θˆ +,θ˜ ∈ b˜ ∩ θˆ +, consider θˆ + n o ( , ˜) = t ∈ IN (i ( )) − (i ( ˜)) Ξ θ θ ; πt,θˆ + t θ πt,θˆ + t θ . β and note that we can write m Nt,θˆ + (β) = ∑ ∑ ∑ χΞ(θ,θ˜)(t). (9.5) b˜∈W− (α,β) k=1 b∈W− (α,β); θˆ+ θˆ+ −k−1 −k 2 ≤Ds(b∧b˜)<2

By Proposition II.27, we have that

β Leb Ξ(θ,θ˜) . . (9.6) ρDs(b ∧ b˜)

Consider m ∈ N such that 8C¯ 2−m = ρ−κ0 β. (9.7) C¯0 ˜ − ˜ −M−1 Also consider M ∈ N, M ≥ m, such that if for b,b ∈ Waˆ (β,ζ), Ds(b∧b) < 2 , then b ⊂ b˜ or b˜ ⊂ b. Note that M − m ρ−κ0 . Observe that

Z M ˜ Nt,θˆ + (β)dt = ∑ ∑ ∑ Leb(Ξ(θ,θ)) b˜∈W− (α,β) k=1 b∈W− (α,β); [− 1 , 1 ]N θˆ+ θˆ+ 2 2 −k−1 −k 2 ≤Ds(b∧b˜)<2

= J1 + J2.

86 We analyze J1 and J2 separately. m ˜ J1 = ∑ ∑ ∑ Leb(Ξ(θ,θ)) b˜∈W− (α,β) k=1 b∈W− (α,β); θˆ+ θˆ+ −k−1 −k 2 ≤Ds(b∧b˜)<2 9.5,PII.20 m −1 k+1 −1 −1 k −d¯s(Λ) . ∑ ∑ ∑ βρ 2 log(A0 β )(2 β) b˜∈W− (α,β) k=1 b∈W− (α,β); θˆ+ θˆ+ −k−1 −k 2 ≤Ds(b∧b˜)<2 m −1 −1 −1 −2d¯s(Λ)+1 k(1−d¯s(Λ)) . log(A0 β )ρ β ∑ 2 k=1 −1 −1 −1 −2d¯s(Λ)+1 m(1−d¯s(Λ)) . log(A0 β )ρ β 2 9.7 −1 −1 −1 −2d¯s(Λ)+1 κ0 −1 1−d¯s(Λ) . log(A0 β )ρ β (ρ β )

−1 −1 −1+κ0(1−d¯s(Λ)) −d¯s(Λ) = log(A0 β )ρ β .

Analogously,

M ˜ J2 = ∑ ∑ ∑ Leb(Ξ(θ,θ)) b˜∈W− (α,β) k=m+1 b∈W− (α,β); θˆ+ θˆ+ −k−1 −k 2 ≤Ds(b∧b˜)<2 κ (1−d¯ (Λ)) −d¯ (Λ) . ρ 0 s β s

−1 −1 −1+κ0(1−d¯s(Λ)) −d¯s(Λ) . log(A0 β )ρ β .

Therefore, Z −1 −1 −1+κ0(1−d¯s(Λ)) −d¯s(Λ) Nt,θˆ + (β)dt . log(A0 β )ρ β . IN

N s ˆ + Fix t ∈ I and choose points ξt,0,...,ξt,k ∈ Wt,loc(θ ) ∩ Haˆ0 , such that

d(ξt,i,ξt,i+1) β for every 0 ≤ i ≤ k − 1. Note that this implies k ζβ −1. − Take b ∈ Waˆ+ (α,β). Consider the following quantities n ˆ + o lt,β (b) = min 0 ≤ j ≤ k; ξt, j ∈ πt,θˆ + (it(b ∩ θ )) and − mt, j(β) = b ∈ Waˆ+ (α,β); lt,β (b) = j .

87 Deﬁne,

Bt,θˆ + (β) = {0 ≤ j ≤ k; mt, j(β) 6= 0}.

Note that Bt,θˆ + (β) counts how many ﬁber of the discretization {ξt,i} intersect − ˆ + boxes b ∈ Waˆ (β,ζ) inside of θ . The purpose of this section is to give a lower bound for the cardinality of Bt,θˆ + (β) showing then that this corresponds to a large portion of the total of boxes b. Observe that

k − Waˆ+ (α,β) = ∑ mt, j(β) = ∑ mt, j(β). j=0 j∈Bt,θˆ+ (β)

Hence, by Cauchy-Schwarz inequality, we have

k −2d¯s(Λ) − 2 β W (α,β) ≤ |B + (β)| m (β) . θˆ + t,θˆ ∑ t, j j=0

≤ |Bt,θˆ + (β)|Nt,θˆ + (β)

The previous arguments ﬁll the proof of the following proposition.

Proposition II.29.

Z 1 ¯ ¯ dt log(A−1β −1)ρ−1+κ0(1−ds(Λ))β ds(Λ). (9.8) N . 0 I Bt,θˆ + (β)

The last preposition is enough to find a good lower bound for the set Bt,θˆ + (β) but as we are interested in the projection of the boxes we define the subset B∗ (β) t,θˆ + of Bt,θˆ + (β) which is the subset with maximum cardinality and satisfies that: for each pair i, j ∈ B∗ (β), i 6= j we have t,θˆ +

d(ξt,i,ξt, j) & 4β.

For anya ˆ ∈ Q set

( ¯ ) ρ1−κ0(1−ds(Λ))β −d L(aˆ) = t ∈ IN; |B∗ (β)| . t,θˆ + & −1 −1 −1 log(ρ )log(A0 β ) Proposition II.30. ! \ Leb L(aˆ) = 1 − oρ (1). aˆ∈Q

88 10 PROOF OF THE MAIN THEOREM

In this section we will follow the notation of the previous sections. − For eacha ˆ ∈ Q consider the subset of Waˆ (β,ζ), Γt,aˆ(β), deﬁned of the following way: for each j ∈ B∗ (β) choose a unique box b ∈ W −(β,ζ) such that t,θˆ + aˆ lt,β (b) = j (this exists by the deﬁnition of Bt,θˆ + (β)). Γt,aˆ(β) is then the set of such b’s.

Figure 10.1: Picking boxes with disjoint projections.

Set \ L = L(aˆ). aˆ∈Q Then, by the previous construction we have:

Proposition II.31. For every t ∈ L

¯ ¯ ∗ ρ1−κ0(1−ds(Λ))β −ds(Λ) • |Γt,aˆ(β)| = |B ˆ + (β)| & −1 −1 −1 ; t,θ log(ρ )log(A0 β )

• For every b,b˜ ∈ Γt,aˆ(β), + ˜ ˆ + πt,θˆ + (it(b ∩ θ )) ∩ πt,θˆ + (it(b ∩ θ )) = ∅. Actually, the projection of this boxes have, at least, distance of order 4β.

89 Takea ˆ ∈ Q. We claim that if α is chosen suitably (remember that for every |a+| + + + a ∈ Q, Ds(σ (a )) α) we have that for every θ ⊂ aˆ and for every pair ˜ b,b ∈ Γt,θˆ + (β) + ˜ + πt,θ + (it(b ∩ θ )) ∩ πt,θ + (it(b ∩ θ )) = ∅. for every t ∈ L. Indeed, assume by contradiction that there exist θ ∈ b ∩ θ + and θ˜ ∈ b˜ ∩ θ + such that ˜ πt,θ + (it(θ)) = πt,θ + (it(θ)). ˆ + u ˜ ˆ + u ˜ Let x ∈ it(b ∩ θ ) ∩Wt,loc(it(θ)) andx ˜ ∈ it(b ∩ θ ) ∩Wt,loc(it(θ)). Note that

−1 [i f ] d(it(θ),x) . d(θ,it (x)) + −|a |[i ] [i f ]/log(τ−1) . 2 f . α

The same holds for d(it(θ˜),x˜). s + + + Remembering that πt,aˆ : ∪θ ⊂a Wt,loc(θ ) → Haˆ0 is η-Holder¨ continuous (the strong stable foliation W ss is Holder¨ continuous) we have, ˜ |πt,θ + (x) − πt,θ +(x˜) ≤ |πt,aˆ(x) − πt,aˆ(it(θ))| + |πt,aˆ(x˜) − πt,aˆ(it(θ))| η ˜ η . d(x,it(θ)) + d(x˜,it(θ)) [i f ] · η/log(τ−1) . 2α .

Assume that

log(τ−1) α = β /η[i f ]. (10.1)

This contradicts the deﬁnition of B∗ (β) and ﬁnishes the claim which is sum- t,θˆ + marized in the next proposition.

log(τ−1) Proposition II.32. Take α = β /η[i f ]. Then, for every a = a− ∩a+ ∈ Q and for every b,b˜ ∈ Γt,a(β) we have that + ˜ + πt,a(it(b ∩ a )) ∩ πt,a(it(b ∩ a )) = ∅, for every t ∈ L.

Consider the number m0 ∈ N with the property that any two letters in A can be connect by a word of size m0 (remember that horseshoe is mixing). Deﬁne the set

+ − + Xt(β) = c · b ∩ a · d; a = a ∩ a ∈ Q, b ∈ Γt,a(β) and |c| = |d| = m0

90 + Since Ds(c·b) β and Ds(a ·d) α we can ﬁnd n,l ∈ N such that ”most” of the the words cb˙ has size l and a+ · d has size n. Indeed, note that if, for example,

Ds(b) β, then log(β −1) log(β −1) |b| log(τ−1) . . log(τˆ−1) which is an interval with size of logarithm order in β. However we have approxi- −d¯ (Λ) mately β s pieces b ∈ Γt,a(β) for every a ∈ Q. So, by the pigeonhole principle we still have exponential quantity of b’s such that |b| is the same. More precisely, −1 −1 there exists n ∈ , log(β ) ≤ n ≤ log(β ) , such that N log(τ−1) log(τˆ−1)

¯ ¯ ρ1−κ0(1−ds(Λ))β −d(Λ) |{b ∈ Γt,a(β); |b| = n}| . & −1 −1 −1 −1 log(ρ )log(β )log(A0 β )

See II.31. The same can be done to ﬁnd l ∈ N. Observe that n,l depends on β. n+l Consider the horseshoe for Ft = ft taking the maximal invariant of a union of disjoint neighborhoods, U(A), that contains the elements it(A) with A ∈ Xt(β). Deﬁne   \ − j [ Λt(β) = Ft  U(A). j∈Z A∈Xt (β)

It is worthwhile to mention that, by the choose of m0 in the deﬁnition of Xt(β), the horseshoe Λt(β) is not trivial. − + Let V ⊂ W be the sets ﬁnite words with alphabet Xt(β) and let V ,V be, respectively, the set of negative and positive words in V.

Proposition II.33. It holds,

1. For every x ∈ Λt(β) ss WFt (x) ∩ Λt(β) = {x};

2. κ0(1 − d¯s(Λ)) − 1 d¯s(Λt(β)) ≥ d¯s(Λ) + − oρ (1), m(1 + κ0) where we choose β = ρm. with m independent of t or ρ.

Proof. Assume, by contradiction, that there exist x ∈ Λ (β) andx ˜ ∈ W ss (x) ∩ t Ft ,loc − Λt(β). Take B,B˜ ∈ V such that x ∈ B andx ˜ ∈ B˜. So, there exist b,b˜ ∈ Γt,a(β), for some a ∈ Q, and c,c˜ of size m0, such that

c · b · B ∧ B˜ = B andc ˜· b˜ · B ∧ B˜ = B˜.

91 This implies, by Proposition II.32 that, if k = |B ∧ B˜|, then

−k + −k ˜ + πt,a(Ft (it(B ∩ a ))) ∩ πt,a(Ft (it(B ∩ a ))) = ∅.

−k −k + −k −k + However, Ft (x) ∈ Ft (it(B∩a )) and Ft (x˜) ∈ Ft (it(B˜ ∩a )) which implies that −k −k πt,a(Ft (x)) = πt,a(Ft (x˜)). Contradiction. So, ss WFt (x) ∩ Λt(β) = {x}. To proof the second part, ﬁrst observe that we can write any A ∈ V−, |A| = M as + + A = c−M+1 · b−M+1 ∩ a−M+1 · d−M+1 ···c0 · b0 ∩ a0 · d0, + − + |a−i| + with |c−i| = m0, a−i = a−i ∩ a−i ∈ Q, Ds(σ (a−i) α and b−i ∈ Γt,a−i (β) for all 0 ≤ i ≤ M, which implies that

D (F ,A) ≥ C˜−3MC−M · M · M s t m0 β α (10.2) ˜ where C > 1 is the constant in the bounded distortion bounds and Cm0 > 1 is a constant depending only on m0. We also note that, for a positive constant C we have

M 1−κ (1−d¯ (Λ)) −d¯ (Λ) ! ρ 0 s β s ¯ A ∈ V−; |A| = M ≥ C · · α−ds(Λ) . −1 −1 −1 −1 log(ρ )log(β )log(A0 β ) (10.3)

By the deﬁnition of λM(Ft) and using inequalities 10.2 and 10.3, we have !M 1−κ0(1−d¯s(Λ)) −d¯s(Λ) −1 MλM(Ft ) ρ β −d¯s(Λ) 1 ≥ ((CC˜ m ) βα) · C · · α 0 −1 −1 −1 −1 log(ρ )log(β )log(A0 β ) which implies that

˜ λM(Ft)log(CCm0 ) ≥ −1 −1 ≥ (d¯s(Λ) − λM(Ft))log((βα) ) + logC + [κ0(1 − d¯d(Λ)) − 1]log(ρ )− −1 −1 −1 −1 − log[log(ρ )log(β )log(A0 β )] ≥

−1 log(τ κ m Write = κ1 and remember that α = β 1 . Also write β = ρ (m will be take η[i f ] large). Then, βα = ρm(1+κ1)

92 So, ˜ λM(Ft)log(CCm0 ) ≥ ¯ k0(1 − ds(Λ)) − 1 −1 ≥ d¯s(Λ) − λM(Ft) + · m(1 + κ1)log(ρ ) + logC− m(1 + κ1) −1 −1 −1 −1 − log[log(ρ )log(β )log(A0 β )]. m+n Since λM(Ft) is bounded (d¯s(Λt(β)) ≤ d¯s( f (Λt) + δ ≤ d¯s(Λ) + δ), dividing −1 everything by m(1 + κ1)log(ρ ) we have that

κ0(1 − d¯s(Λ)) − 1 oρ (1) ≥ d¯s(Λ) − λM(Ft) + . m(1 + κ1)

Given ε > 0 take consider κ (1 − d¯ (Λ)) − 1 m ≥ 0 s . ε(1 + κ1) So, by Proposition II.33, and for all t ∈ L

d¯s(Λt(β)) ≥ d¯s(Λ) − oρ (1) + ε, and for every x ∈ Λt(β) ss WFt (x) ∩ Λt(β) = {x}. j Observe that ft (Λt(β)) are subsets invariant by Ft, for all 1 ≤ j ≤ n + l − 1, and have the same properties as Λt(β), above described. 0 Using the Proposition II.22 we extract from Λt(β) a subhorseshoe Λt(ρ) such that

• For all i, j ∈ {0,...,m + n − 1} and for all i 6= j.

j 0 i 0 ft (Λt(ρ)) ∩ ft (Λt(ρ)) = ∅;

¯ 0 ¯ • ds(Λt(ρ)) ≥ ds(Λt(β) − ε for all i, j ∈ {0,...,m + n − 1} and for all i 6= j. The horseshoe n+l−1 [ j 0 ft (Λt(ρ)) j=0 satisﬁes the property of the Main theorem for every t ∈ L. Since L has positive measure (actually a measure close to 1) in particular is not empty. This concludes the proof of the Main theorem.

93 BIBLIOGRAPHY

[1] A. Avila, J. Santamaria, and M. Viana. Holonomy invariance: rough regularity and applications to Lyapunov exponents. Asterisque´ , 358:13–74, 2013.

[2] A. Avila and M. Viana. Simplicity of Lyapunov spectra: a sufﬁcient crite- rion. Port. Math., 64:311–376, 2007.

[3] A. Avila and M. Viana. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math., 181(1):115–189, 2010.

[4] Artur Avila. Density of positive Lyapunov exponents for sl(2,r)-cocycles. J. Amer. Math. Soc., 24:999–1014, 2011.

[5] L. Backes, M. Poletti, and A. Sanchez.´ The set of ﬁber-bunched cocyles with nonvanishing lyapunov exponents over a partially hyperbolic map is open. Mathematical Research Letters, 25:1719–1740, 2018.

[6] L. Backes, M. Poletti, P. Varandas, and Y. Lima. Simplicity of lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems. Er- godic Theory and Dynamical Systems, page 1–23, 2019.

[7] Alex Blumenthal, Jinxin Xue, and Lai-Sang Young. Lyapunov exponents for random perturbations of some area-preserving maps including the standard map. Ann. of Math. (2), 185(1):285–310, 2017.

[8] J. Bochi. Genericity of zero Lyapunov exponents. Ergod. Th. & Dynam. Sys., 22:1667–1696, 2002.

[9] Jacek Bochnak, Michel Coste, and Marie-Franc¸oise Roy. Real algebraic geometry, volume 36. Springer Science & Business Media, 2013.

[10] C. Bonatti and M. Viana. Lyapunov exponents with multiplicity 1 for deterministic products of matrices. Ergod. Th. & Dynam. Sys, 24:1295–1330, 2004.

94 [11] Christian Bonatti and Sylvain Crovisier. Center manifolds for partially hyperbolic sets without strong unstable connections. Journal of the Institute of Mathematics of Jussieu, 15(4):785–828, 2016.

[12] Christian Bonatti, Sylvain Crovisier, Lorenzo Diaz, and Amie Wilkinson. What is... a blender? arXiv preprint arXiv:1608.02848, 2016.

[13] Christian Bonatti, Lorenzo Justiniano D´ıaz, and Marcelo Viana. Disconti- nuity of the hausdorff dimension of hyperbolic sets. Comptes Rendus de l’Academie des Sciences-Serie I-Mathematique, 320(6):713–718, 1995.

[14] Rufus Bowen. Some systems with unique equilibrium states. Mathematical systems theory, 8(3):193–202, 1974.

[15] Rufus Bowen. Equilibrium states and the ergodic theory of anosov diffeomorphisms. Springer Lecture Notes in Math, 470:78–104, 1975.

[16] Sylvain Crovisier and Rafael Potrie. Introduction to partially hyperbolic dynamics. School on Dynamical Systems, ICTP, Trieste, 3:1, 2015.

[17] David Damanik. Schrodinger¨ operators with dynamically deﬁned potentials. Ergodic Theory Dynam. Systems, 37(6):1681–1764, 2017.

[18] P. Duarte and S. Klein. Continuity positivity and simplicity of the Lyapunov exponents for linear quasi-periodic cocycles. Preprint https://arxiv.org/pdf/1603.06851.pdf.

[19] Paul Erdos.¨ Graph theory and probability. Canadian Journal of Mathemat- ics, 11:34–38, 1959.

[20] K. Falconer. Fractal geometry. John Wiley & Sons Ltd., 1990. Mathematical foundations and applications.

[21] John Franks. Necessary conditions for stability of diffeomorphisms. Trans- actions of the American Mathematical Society, 158(2):301–308, 1971.

[22] I. Ya. Gol’dsheid and G. A. Margulis. Lyapunov indices of a product of random matrices. Uspekhi Mat. Nauk., 44:13–60, 1989.

[23] M Golubitsky and V Guillemin. Stable mappings and their singularities. Springer-Verlag New York Inc., 1973.

95 [24] Y. Guivarc’h and A. Raugi. Products of random matrices : convergence theorems. Contemp. Math., 50:31–54, 1986.

[25] Morris W Hirsch, Charles C Pugh, and Michael Shub. Invariant manifolds. Bulletin of the American Mathematical Society, 76(5):1015–1019, 1970.

[26] Anatole Katok and Boris Hasselblatt. Introduction to the modern theory of dynamical systems, volume 54. Cambridge university press, 1997.

[27] Franc¸ois Ledrappier, Michael Shub, Carles Simo,´ and Amie Wilkinson. Ran- dom versus deterministic exponents in a rich family of diffeomorphisms. J. Statist. Phys., 113(1-2):85–149, 2003.

[28] Carlos Matheus, Carlos Gustavo Moreira, and Waliston Luiz Silva. Geome- try of large horseshoes in 3-manifolds. In preparation, 2020.

[29] Carlos Gustavo Moreira, Jacob Palis, and Marcelo Viana. Homoclinic tangencies and fractal invariants in arbitrary dimension. Comptes Rendus de l’Academie´ des Sciences-Series I-Mathematics, 333(5):475–480, 2001.

[30] Carlos Gustavo Moreira and Jean-Christophe Yoccoz. Tangences homo- clines stables pour des ensembles hyperboliques de grande dimension frac- tale. In Annales scientiﬁques de l’Ecole normale superieure´ , volume 43, pages 1–68, 2010.

[31] Carlos Gustavo T de A Moreira. Stable intersections of cantor sets and homoclinic bifurcations. In Annales de l’Institut Henri Poincare (C) Non Lin- ear Analysis, volume 13, pages 741–781. Elsevier, 1996.

[32] Carlos Gustavo T de A Moreira and Jean-Christophe Yoccoz. Corrigendum: Stable intersections of regular cantor sets with large hausdorff dimensions. Annals of Mathematics, 154(2), 2001.

[33] S. Newhouse. Diffeomorphisms with inﬁnitely many sinks. Topology, 13:9– 18, 1974.

[34] Sheldon Newhouse. Non-density of axiom a (a) on s2, global analysis. In Proceedings of Symposia in Pure Mathematics, volume 14, pages 191–202, 1968.

[35] Sheldon E Newhouse, Jacob Palis, and Floris Takens. Bifurcations and stability of families of diffeomorphisms. Publications Mathematiques´ de l’IHES´ , 57:5–71, 1983.

96 [36] Davi Obata and Mauricio Poletti. On the genericity of positive exponents of conservative skew products with two-dimensional ﬁbers. arXiv preprint arXiv:1809.03874.

[37] V. I. Oseledets. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 19:197–231, 1968.

[38] Jacob Palis. Homoclinic orbits, hyperbolic dynamics and dimension of cantor sets. Contemp. Math, 58(26):203–216, 1987.

[39] Jacob Palis and Floris Takens. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: Fractal dimensions and inﬁnitely many attractors in dynamics, volume 35. Cambridge University Press, 1995.

[40] Jacob Palis and Marcelo Viana. High dimension diffeomorphisms displaying inﬁnitely many periodic attractors. Annals of mathematics, pages 207–250, 1994.

[41] Henri Poincare´ and R Magini. Les methodes´ nouvelles de la mecanique´ celeste.´ Il Nuovo Cimento (1895-1900), 10(1):128–130, 1899.

[42] M. Poletti. Stably positive lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems - A, 38:5163, 2018.

[43] M. Poletti and M. Viana. Simple lyapunov spectrum for certain linear cocycles over partially hyperbolic maps. Nonlinearity, 32(1):238, 2018.

[44] Charles Pugh, Michael Shub, Amie Wilkinson, et al. Holder¨ foliations. Duke Mathematical Journal, 86(3):517–546, 1997.

[45] Charles Pugh, Michael Shub, Amie Wilkinson, et al. Correction to:” holder¨ foliations” duke math. j. vol. 86 no. 3 (1997), pp. 517-546. Duke Mathemat- ical Journal, 105(1):105–106, 2000.

[46] Michael Shub. Global stability of dynamical systems. Springer Science & Business Media, 2013.

[47] M. Viana. Multidimensional nonhyperbolic attractors. Inst. Hautes Etudes´ Sci. Publ. Math., 85:63–96, 1997.

97 [48] M. Viana. Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. Ann. of Math., 167:643–680, 2008.

[49] Marcelo Viana and Krerley Oliveira. Foundations of ergodic theory, volume 151. Cambridge University Press, 2016.

[50] Y. Wang and J. You. Quasi-periodic Schrodinger¨ cocycles with positive Lyapunov exponent are not open in the smooth topology. Preprint arXiv:1501.05380, 2015.