ON THE NUMERICAL CONSTRUCTION OF HYPERBOLIC STRUCTURES FOR COMPLEX DYNAMICAL SYSTEMS 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 Jennifer Suzanne Lynch Hruska August 2002 c 2002 Jennifer Suzanne Lynch Hruska ALL RIGHTS RESERVED ON THE NUMERICAL CONSTRUCTION OF HYPERBOLIC STRUCTURES FOR COMPLEX DYNAMICAL SYSTEMS Jennifer Suzanne Lynch Hruska, Ph.D. Cornell University 2002 Our main interest is using a computer to rigorously study -pseudo orbits for polynomial diffeomorphisms of C2. Periodic -pseudo orbits form the -chain re- current set, R. The intersection ∩>0R is the chain recurrent set, R. This set is of fundamental importance in dynamical systems. Due to the theoretical and practical difficulties involved in the study of C2, computers will presumably play a role in such efforts. Our aim is to use computers not only for inspiration, but to perform rigorous mathematical proofs. In this dissertation, we develop a computer program, called Hypatia, which locates R, sorts points into components according to their -dynamics, and inves- tigates the property of hyperbolicity on R. The output is either “yes”, in which case the computation proves hyperbolicity, or “not for this ”, in which case infor- mation is provided on numerical or dynamical obstructions. A diffeomorphism f is hyperbolic on a set X if for each x there is a splitting of the tangent bundle of x into an unstable and a stable direction, with the unstable (stable) direction expanded by f (f −1). A diffeomorphism is hyperbolic if it is hyperbolic on its chain recurrent set. Hyperbolicity is an interesting property for several reasons. Hyperbolic diffeo- morphisms exhibit shadowing on R, i.e., -pseudo orbits are δ-close to true orbits. Thus they can be understood using combinatorial models. Shadowing also implies structural stablity, i.e., in a neighborhood in parameter space the behavior is con- stant. These properties make hyperbolic diffeomorphisms amenable to computer investigation via -pseudo orbits. We first discuss Hypatia for polynomial maps of C. We then extend to polyno- mial diffeomorphisms of C2. In particular, we examine the class of H´enondiffeo- morphisms, given by 2 Ha,c :(x, y) → (x + c − ay, x). This is a large class of diffeomorphisms which provide a good starting point for understanding polynomial diffeomorphisms of C2. However, basic questions about the complex H´enon family remain unanswered. In this work, we describe some H´enondiffeomorphisms for which Hypatia ver- ifies hyperbolicity, and the obstructions found in testing hyperbolicity of other examples. BIOGRAPHICAL SKETCH Jennifer Suzanne Lynch Hruska moved to Ithaca, New York, in May 1997. Here she lived, for the first time, away from her home state of Missouri, away from her parents, sisters, grandparents, great grandmother, aunts, uncles and cousins. Suzanne had just completed her B.S. in applied mathematics from the University of Missouri at Rolla, only 30 miles from her childhood town of Waynesville. In Ithaca she attended Cornell University with the intention of earning her Ph.D. in mathematics, an intention which has been realized with the submission of this dis- sertation. She agrees with Cornell’s Director of Graduate Studies in mathematics, who explained at the May 2002 commencement exercises that graduate students are to learn, to learn how to learn, to learn how to impart knowledge to others, and to learn how to create knowledge. Cornell and Ithaca have provided an excellent environment for Suzanne to achieve these goals. During this process, she also earned her M.S. in mathematics from Cornell (August 1999) and met and married fellow mathematics graduate student Geoffrey Christopher Hruska (June 2000). She became a vegetarian and learned to cook, swing dance and program a computer. She traveled to Barcelona, Marseilles and Paris. In August 2002, Suzanne will move to Long Island to be a (VIGRE) postdoc- toral fellow at the State University of New York at Stony Brook. iii For Terry and Jan Primas iv ACKNOWLEDGEMENTS During my five years at Cornell, many people have provided encouragement and support. Friends, family, and colleagues have all contributed to my finishing this process with my sanity intact. I would like to individually acknowledge a few who stand out as having been particularly helpful. My advisor, John Smillie, has patiently shared his knowledge and wisdom during many many meetings. He has provided sound guidance in each phase of my studies, from the formation of my dissertation topic to securing a good position for after I leave Cornell. John Hamal Hubbard’s enthusiasm, apparent on my first visit to Cornell as a prospective grad- uate student, brought me to the subject of holomorphic dynamical systems. His suggestions led to this dissertation topic and have been useful for studying the problem. Greg Buzzard helped me to gain an improved perspective on mathemat- ics and life any time I was starting to feel overwhelmed by the process. We had many useful discussions when I was first reading papers on the H´enon family, and in the early stages of my dissertation. John Guckenheimer provided feedback on this dissertation and served on my special committee, and Clifford Earle intro- duced me to complex analysis in the graduate course he taught my first year at Cornell, as well as served as a member of my special committee. I also had helpful conversations with Yulij Ilyashenko, Eric Bedford, John Erik Fornaess, Mattias Jonsson, Ricardo Oliva, and Roland Roeder. I would also like to thank Adrien Douady, for invitations and hospitality. Harnessing the power of a computer has been a central theme in my work. Warwick Tucker introduced me to a key technique in rigorous computing, interval arithmetic. Karl Papadantonakis, along with other members of Cornell’s “H´enon working group” (Smillie, Hubbard, Guckenheimer, Ilyashenko, Buzzard, Tucker, Ricardo Oliva, David Brown, Yutaka Ishii, and occasionally Bedford, Douady, and Peter Papadopol), opened a world of fascinating possibilities in the dynamics of the H´enonfamily through the development and examination of new computer programs FractalAsm and SaddleDrop ([1]). Additionally, Bob Terrell has often provided technical support in his capacity as System Administrator for the Cornell mathematics department’s excellent computing facilities. My work benefited from the use of computers purchased with funds from a SCREMS grant from the NSF (National Science Foundation). As a graduate student, I received financial support through a variety of sources, most notably Cornell University and the NSF. Various sources of funds supported my participation in conferences and allowed increased time for working on math- ematical research. Graduate school is extremely rewarding overall, but at times can be incredibly frustrating. My parents, my sisters, my closest friends, and most of all my husband deserve high praise for not only putting up with me these five years, but making my time here better in myriad ways. v vi TABLE OF CONTENTS 1 Introduction and Background 1 1.1 The H´enonfamily . 1 1.2 Invariant sets of interest . 2 1.3 Hyperbolicity . 8 1.4 Overview of this dissertation: Hypatia ................. 19 2 The box-chain recurrent set 21 2.1 Trapping regions for Rδ ......................... 21 2.2 The box-chain recurrent graph . 23 2.3 Algorithm to build a box-chain recurrent graph . 28 3 Hyperbolicity in one complex dimension 31 3.1 Introduction: box-expansion . 31 3.2 Algorithm to build a metric showing box-expansion . 32 3.3 Box-expansion implies continuous expansion . 34 3.4 Finding a good expansion amount . 35 3.4.1 An optimal, yet impractical solution . 36 3.4.2 An approximate, efficient solution . 38 3.5 Building a better metric . 39 3.5.1 An optimal, yet very impractical solution . 40 3.5.2 An approximate, realistic solution . 44 3.6 Examples of running Hypatia for polynomial maps . 47 4 Hyperbolicity in two complex dimensions 65 4.1 Introduction: box-hyperbolicity . 65 4.2 Algorithm to show box-hyperbolicity . 66 4.2.1 Defining stable and unstable directions . 66 4.2.2 Defining a metric on the directions . 68 4.2.3 Defining a cone field . 70 4.3 Characterizing box-hyperbolicity . 71 4.3.1 Checking box-hyperbolicity with Hermitian forms . 71 4.3.2 Box-hyperbolicity implies hyperbolicity . 73 4.4 Examples of running Hypatia for H´enondiffeomorphisms . 77 4.4.1 A generic run of Hypatia for a H´enondiffeomorphism . 77 4.4.2 Box-hyperbolic H´enon diffeomorphisms . 86 4.4.3 Describing box covers which are not box-hyperbolic . 86 5 Analysis of Hypatia’s performance 97 5.1 Separating J from a fixed sink . 97 5.2 Conclusions . 107 A Interval arithmetic 109 vii B Pseudo-code 111 References 114 viii LIST OF TABLES 2 3.1 Examples of running Hypatia for Pc(x) = x + c........... 55 2 2 3.2 Examples of running Hypatia for Pc,a(z) = z − 3a z + c....... 64 4.1 Data from verifying hyperbolicity with Hypatia............ 87 4.2 Data on subdivision for Example 4.4.4. 88 4.3 Data on stable/unstable directions for Example 4.4.4. 89 4.4 Data on first metric and cones for Example 4.4.4. 90 4.5 Data on subdivision for Example 4.4.5. 92 4.6 Data on stable/unstable directions for Example 4.4.5. 93 4.7 Data on subdivision for Example 4.4.6. 94 4.8 Data on stable/unstable directions for Example 4.4.6. 95 4.9 Data comparing Examples 4.4.5 and 4.4.6. 96 5.1 Constants for sink/J separation estimates for Example 5.1.1. 107 ix LIST OF FIGURES 1.1 -recurrence .