ON THE COMBINATORICS OF EXTERNAL RAYS IN THE DYNAMICS OF THE COMPLEX HENON MAP. A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Ricardo Antonio Oliva May 1998 c Ricardo Antonio Oliva 1998 ALL RIGHTS RESERVED Addendum. This is a slightly revised version of my doctoral thesis: some typing and spelling mistakes have been corrected and a few sentences have been re-worded for better legibility (particularly in section 4.3). Also, to create a nicer pdf document with hyperref, the title of section 3.3.2 has been made shorter. The original title was A model for a map with an attracting fixed point as well as a period-3 sink: the (3-1)-graph. ON THE COMBINATORICS OF EXTERNAL RAYS IN THE DYNAMICS OF THE COMPLEX HENON MAP. Ricardo Antonio Oliva , Ph.D. Cornell University 1998 We present combinatorial models that describe quotients of the solenoid arising from the dynamics of the complex H´enon map 2 2 2 fa,c : C → C , (x, y) → (x + c − ay, x). These models encode identifications of external rays for specific mappings in the H´enon family. We investigate the structure of a region of parameter space in R2 empirically, using computational tools we developed for this study. We give a combi- natorial description of bifurcations arising from changes in the set of identifications of external rays. Our techniques enable us to detect, predict, and locate bifurca- tion curves in parameter space. We describe a specific family of bifurcations in a region of real parameter space for which the mappings were expected to have sim- ple dynamics. We compute the first few bifurcation curves in this family and label them combinatorially. Our computer experiments also indicate the existence of gaps within the region of real parameter space where H´enon family fa,c has connected Julia set Ja,c. We show why the verification of this gap would imply the existence of values of a for which the level-a Mandelbrot set, Ma = {c ∈ C : Ja,c is connected}, is not connected. BIOGRAPHICAL SKETCH Ricardo Oliva was born in Valpara´ıso, Chile, where he hopes to return some day. He came to the US with his parents just before entering high school; three years later he moved to Ithaca, NY, to study at Cornell University where he has been for nearly half of his life and where he also hopes to return some day. At Cornell Ricardo also completed B.S. degree in Applied & Engineering Physics and a M.S. degree in Computer Science. After leaving Cornell Ricardo will spend a year at the Institute for Mathematics and its Applications, University of Minnesota, in the program on Emerging Applications of Dynamical Systems. Since the Spring equinox of 1986 Ricardo has shared his life with his wife Marta del Campo who has been in his heart since 4th grade of elementary school, and who also will be receiving her Doctorate from Cornell this year; their son Manuel was born two months after Ricardo entered graduate school and already has earned a “doctorate in patience”. iii AManuel, por todos esos d´ıas que no pateamos la pelota. A Marta, por todo lo que esto nos cost´o. Y a la memoria de Aragrev, con qui´en yo esperaba compartir estas locuras pero no alcanc´eadec´ırcelo. Lo m´as terrible se aprende enseguida y lo hermoso nos cuesta la vida. – Silvio Rodr´ıguez. iv ACKNOWLEDGEMENTS I would like to begin by thanking my advisor, Professor John Smillie, for his guidance and encouragement throughout the duration of this project, and for his generous support during my later years as a graduate student. This thesis is also a product of his efforts and not just in an indirect way. We held many discussions in which not only he shared his ideas but also he helped focus mine; I am grateful for his dedication as a mentor. Much of my gratitude goes also to Professor John Hubbard, who almost exactly four years ago showed me a picture of an unstable manifold for a complex H´enon map and invited me into the problem of studying the combinatorics of these mappings. His enthusiasm for this subject and his initial guidance were pivotal in my choice of topic. It was in his office and his Macintosh where I began the investigations which develop into this thesis, and I thank him for the many ideas he shared with me. Also I would like to thank Professor John Guckenheimer who welcomed me as an extra student at Cornell’s Center of Applied Math. The excellent computer facilities at CAM facilitated the the computer investigations on which much of this work is based. v The Cornell Theory Center grants DMS931236C and DMS980217C provided me with access to the SP2 supercomputer where we produced most of the pictures used in this study. I am also grateful to to Professor Eric Bedford of Indiana University for helpful discussions while I worked in this project. His questions and suggestions led us to consider the idea of scanning parameter space to detect regions of stability. Bernd Krauskopf got us thinking about ways in which bifurcations curves could be traced in parameter space. Both ideas played a central role in this project. During my initial years as a graduate student I benefited from the advice and encouragement of many people. In particular, I would like to thank Professor Mar- shall Cohen who served in my original graduate student Committee; Tom Rishell who taught me my first serious math course when I was still an engineering student and who motivated me to pursue graduate studies in Mathematics; and Ernesto Acosta who took time from his busy last year as a graduate student to answer the many math questions I asked him during my first year. Finally, to my family and friends, thanks for always being there, especially at the end when things got tough in so many ways. vi Table of Contents 1 Introduction 1 1.1 Thesis Summary. ............................ 4 2 Background and Preliminaries 6 2.1 External rays for polynomials on C. .................. 6 2.2 Complex H´enon Mappings and their Julia set. ............ 7 2.3 The complex solenoid .......................... 10 2.3.1 External rays on the complex solenoid. ............ 12 2.3.2 Parameterization of components. ............... 13 2.3.3 Binary Representation of Σ0 .................. 16 − 2.4 The solenoid as a model for J+ . .................... 23 2.4.1 Hyperbolic maps with connected J. .............. 25 2.5 The unstable manifold “picture” of a periodic saddle. ........ 27 u 2.5.1 Reading the Wp picture. .................... 31 3 Combinatorial models for H´enon Mappings 37 3.1 Graph representation of quotients of S2. ............... 37 3.2 Perturbations of hyperbolic polynomials. ............... 43 3.3 Two-dimensional examples. ...................... 52 3.3.1 A H´enon map with period-2 sink. ............... 52 3.3.2 A model for a map with sinks of period 1 and 3 ....... 57 3.3.3 Verification of the planarity conditions ............ 60 4 On the Combinatorial Structure of a Region of Real Parameter Space 80 4.1 Introduction. .............................. 80 4.1.1 Scanning Parameter Space. .................. 87 4.2 Locating bifurcations in parameter space. ............... 96 4.2.1 Some specific bifurcation curves in Region 2. ......... 100 4.2.2 Describing all identifications in a stable region. ........ 103 4.3 Bifurcations predicted by the models. ................. 111 vii 4.3.1 Computing P (h,n), ... ,S(h,n), and P (0), ... ,S(0). ..... 113 4.3.2 A Cantor fan of bifurcation lines. ............... 118 4.4 Disconnectivity of the connectivity locus. ............... 123 Appendix 127 A Other bifurcation curves above region-2. ............... 127 B Notes on the algorithms. ........................ 134 B.1 Computing Unstable Manifolds. ................ 134 B.2 Computing G+ and its harmonic conjugate. ......... 135 B.3 Computing External Rays ................... 136 Bibliography 139 viii List of Tables 3.1 Possible [n0,...,n2]-block of four sequences A < B < A < B subject to the conditions that exactly k of the sequences agree on all coordinates less than n4−k, k =4, 3, 2. The entries x and y denote words and ∗ is a “wild-card” (of the right length in each column) representing arbitrary entries. .................................. 64 3.2 Possible [n0,...,n2]-block in four sequences A < B < A < B which agree on all coordinates less than n0 and such that no three of the sequences agree on the n0 coordinate. The coordinates n1 and n2 correspond to the first distinct entry between each pair of sequences that agreed on the n0 coordinate. ............................... 65 ix List of Figures 2.1 ..................................... 9 2.2 ..................................... 30 + 2.3 A fundamental domain in Uβ . .................... 33 u 2.4 Wβ picture with four real identifications in a fundamental domain. ... 36 3.1 The diagonal graph. 38 3.2 The solenoid graph ΓΣ0 ......................... 39 3.3 The graph Γ2. .............................. 41 3.4 The set Kc for c = −1.......................... 42 3.5 ..................................... 49 3.6 ..................................... 50 u − 3.7 Wa,c picture for a =0.125,c= 1.124 .................. 53 3.8 The graph Γ2. .............................. 55 3.9 ..................................... 56 u − 3.10 Wa,c picture for a =0.3,c= 1.17 ................... 58 3.11 The graph Γ(3,1). ............................ 59 3.12 The graph Γ2 with marked edges referenced in the proof of Lemma 21. 68 3.13 The graph Γ(3,1) with marked edges referenced in the proof of Lemma 22. 72 4.1 M and its hyperbolic components of period 1, 2 and 3. ......... 82 4.2 Sink loci of fa,c in c-plane for small values of a.