Instituto Nacional de Matemática Pura e Aplicada Doctoral Thesis KNEADING SEQUENCES FOR TOY MODELS OF HÉNON MAPS Ermerson Rocha Araujo Rio de Janeiro July 30, 2018 Instituto Nacional de Matemática Pura e Aplicada Ermerson Rocha Araujo KNEADING SEQUENCES FOR TOY MODELS OF HÉNON MAPS Thesis presented to the Post-graduate Program in Math- ematics at Instituto Nacional de Matemática Pura e Aplicada as partial fulfillment of the requirements for the degree of Doctor in Philosophy in Mathematics. Advisor: Enrique Ramiro Pujals Rio de Janeiro 2018 Amar e mudar as coisas me interessa mais. Alucinação - Belchior To Sheila Cristina and José Ribamar Acknowledgments I would like to especially express my gratitude to my advisor Enrique Pujals. I thank him for constant encouragement and for many insightful conversa- tions during the preparation of the thesis. Our meetings were almost always held in the IMPA’s restaurant. Which was very good for me, because this took away the pressure that the elaboration of a thesis puts on the students. At each meeting with him many new ideas arose for the problems we were working on. He made extremely complicated arguments seem simple. Besides being an extraordinary mathematician, he is a great person, generous and very attentive. So I can say that it was an honor and an immense adventure to have been a Pujals’ student. I thank IMPA for these four years in which I had the opportunity to grow as a mathematician. In addition, I would like to express my gratitude to the Graduate Center-CUNY in New York City, USA, where I have been twice. Here I thank Pujals again for his help in making possible my visit at the Graduate Center. I thank especially to Adriana Sánchez and Yaya Tall for many discussions during our journey at IMPA as well as by your friendships, particularly dur- ing our preparation for the Qualification Exam. We are the Trinity of the Dynamics-2014. I would like to thank Alex Zamudio, Cayo Dória, Fernando Lenarduzzi, Pedro Gaspar and Sandoel Vieira for many inspiring discussions and in par- ticular spending many hours with me during the final part of the preparation of the thesis. Especially, I thank Alex Zamudio again for his detailed read- v Ermerson Rocha Araujo Kneading Sequences ing and suggestions for early versions of the thesis that helped me fix small mistakes. I thank my committee members, Alex Zamudio, Carlos Gustavo Moreira (Gugu), Jacob Palis, Patrícia Cirilo, Pierre Arnoux and Yuri Lima, for their comments and suggestions. In Rio de Janeiro, Brazil, I had the company of excellent people. To all, my thanks. Finally, I wish to thank my parents Sheila Cristina and José Ribamar and my brother Herbert Araujo for the support, encouragement and love. Without them the walk would certainly be more difficult. I would like to thank CNPq for giving me the financial support. IMPA vi July, 2018 Abstract We present a study on how a certain type of combinatorial equivalence implies topological conjugacy. In this work, we present the concept of kneading se- quences for a more general setting than one-dimensional dynamics. For this, we consider a two-dimensional family introduced by Benedicks and Carleson [BC91] as a toy model for Hénon maps and define the notion of kneading se- quences of the critical line for a toy model. We show that these sequences are a complete invariant of the conjugacy class of the toy model. Furthermore, we show a version of Singer’s Theorem for toy models and a combinatorial equivalence result for nonautonomous discrete dynamical systems. Keywords: Toy models, kneading sequences, negative Schwarzian derivative, nonautonomous discrete dynamical systems vii Resumo Nós estudamos condições para que uma equivalêcia combinatória implique uma conjugação topológica. Neste trabalho, apresentamos a noção de se- quências kneading para um contexto mais geral em relação a dinâmicas unidi- mensionais. Para este fim, consideramos uma família bidimensional definida por Benedicks e Carleson [BC91] como um “toy model” para aplicações de Hénon e definimos sequências kneading da linha crítica. Mostramos que tais sequências constituem um invariante completo para as classes de conjugação dos “toy models”. Além disso, mostramos um resultado análogo ao Teorema de Singer para nossa família bem como uma equivalência combinatória para sistemas dinâmicos não autônomos. Palavras-chave: Toy models, sequências kneading, derivada Schwarziana negativa, sistemas dinâmicos não autônomos viii Contents Acknowledgments . .v Abstract . vii Resumo . viii 1 Introduction 1 2 The one-dimensional case revisited 6 2.1 The continuous case . .6 2.2 The discontinuous case . 11 2.2.1 One-side topology . 12 2.2.2 Kneading sequence for discontinuous unimodal maps . 12 3 Toy models and kneading sequences 14 3.1 Notations and preliminaries . 15 3.2 Kneading sequences and main results . 20 4 Proofs of the main results 24 4.1 Proof of Theorem 2 . 24 4.2 Proof of Theorem A . 28 4.3 Proof of Theorem C . 32 5 On Singer’s Theorem for toy models 37 6 Open questions and future work 41 6.1 Toy model and further results . 41 ix 6.2 Looking for combinatorial structure for two dimensional dissi- pative diffeomorphisms . 42 A Generalized toy models 44 B Combinatorial equivalence for nonautonomous discrete dy- namical systems 47 B.1 Combinatorial equivalence for NDS . 48 CHAPTER 1 Introduction One of the main questions in Dynamical Systems is whether two systems are ‘the same’, where by ‘the same’ we mean some type of equivalence between two systems. In this sense, one of the most simple ways to say that two systems are equivalent is by obtaining an orbit equivalence map between them. Roughly speaking, we say that two topological dynamical systems are topologically orbit equivalent if there exists a homeomorphism between their phase spaces that preserves their structures and induces a one-to-one correspondence be- tween their orbits. Formally, let X and Y be two topological spaces, and let f : X ! X and g : Y ! Y be functions. We say that f and g are orbit equivalent whenever there is a homeomorphism h : X ! Y sending orbits to orbits, that is, h(Of (x)) = Og(h(x)) for every x 2 X. When f and g are homeomorphisms, the definition means that there exist func- tions α; β : X ! Z such that for all x 2 X, h ◦ f(x) = gα(x) ◦ h(x) and h ◦ f β(x)(x) = g ◦ h(x). So, given f : X ! X a system we want to charac- terize the orbit equivalence class of f, that is, we want to determine the set [f] := fg : X ! X; g is orbit equivalent to fg. The notion of orbit equivalence was firstly studied in the context of prob- ability measure preserving group actions, where the homeomorphism h is replaced by a measurable isomorphism. It follows from works of Dye [Dye59] and [Dye63], Ornstein and Weiss [OW80], and Connes, Feldman and Weiss 1 Ermerson Rocha Araujo Kneading Sequences [CFW81] one of the most remarkable results: Any probability measure pre- serving action of an amenable group is orbit equivalent to a probability mea- sure preserving action of Z. This implies that in the measurable setting there is only one orbit equivalence class, at least when the action is made by an amenable group. One notion stronger than orbit equivalence and which is the main problem addressed in the thesis is topological conjugacy. Let X and Y be topological spaces, and let f : X ! X and g : Y ! Y be continuous functions. We say that f : X ! X and g : Y ! Y are topologically conjugate (or conjugate for simple) if there exists a homeomorphism h : X ! Y satisfying the conjugacy equation h ◦ f = g ◦ h. Note that if f is topologically conjugate to g, then f is orbit equivalent to g. The space where the characterization of the topological conjugacy class was firstly constructed is the circle. In the late XIX century, Poincaré intro- duced the notion of rotation number for orientation preserving homeomor- phisms of circle. He showed that if f : S1 ! S1 is an orientation preserving homeomorphism with irrational rotation number, then there exists a rigid rotation R with the same rotation number such that f is topologically semi- conjugate to R. Furthermore, if f contains a point x whose orbit is dense in S1, then f is topologically conjugate to R. The next step is to consider an interval instead of the circle. However, if f : I ! I is a homeomorphism, then the dynamic of f is trivial, because for all points x 2 I we get that either x is a periodic point or f n(x) converges to a periodic orbit. Therefore, for the interval the class of endomorphisms is more interesting to work with. In this direction, Milnor and Thurston, in their famous paper [MT88], gave origin to the Kneading Theory for endomorphisms of the interval, which is an analogue to the Poincaré theory for homeomorphisms of the circle. More specifically, they considered maps f : I ! I, where I is an interval, with a finite number of turning points (a point of change of monotonicity) and defined the notion of kneading sequences (the itinerary of the turning points). They proved the following theorem: Theorem 1.1. Suppose that f; g : I ! I are two l-modal maps with turning f f g g points c1 < ··· < cl and c1 < ··· < cl . Assume that f and g have no wandering intervals, no intervals of periodic points and no attracting periodic points. If f and g have the same kneading sequences, then f and g are topologically conjugate. IMPA 2 July, 2018 Ermerson Rocha Araujo Kneading Sequences As our main result uses the same ideas, we will give the proof of Theorem 1.1 in the next section.