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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 with- out 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 jour- ney 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 Gonc¸alves, 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 dy- namics) and geometric properties of horseshoes (uniformly hyperbolic dynam- ics). 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 ran- dom 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 dimen- sional 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 ) = fHt; t 2 S g........ 28 0 5.1 Lg 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 z............... 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 maxi- mum 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 ma- trices 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 dimen- sional 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, more- over, 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 cocy- cles 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 deli- cate, 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.
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