ABSTRACT The Specification Property and Chaos in Multidimensional Shift Spaces and General Compact Metric Spaces Reeve Hunter, Ph.D. Advisor: Brian E. Raines, D.Phil. Rufus Bowen introduced the specification property for maps on a compact met- ric space. In this dissertation, we consider some implications of the specification d property for Zd-actions on subshifts of ΣZ as well as on a general compact metric space. In particular, we show that if σ : X X is a continuous Zd-action with ! d a weak form of the specification property on a d-dimensional subshift of ΣZ , then σ exhibits both !-chaos, introduced by Li, and uniform distributional chaos, intro- duced by Schweizer and Smítal. The !-chaos result is further generalized for some broader, directional notions of limit sets and general compact metric spaces with uniform expansion at a fixed point. The Specification Property and Chaos in Multidimensional Shift Spaces and General Compact Metric Spaces by Reeve Hunter, B.A. A Dissertation Approved by the Department of Mathematics Lance L. Littlejohn, Ph.D., Chairperson Submitted to the Graduate Faculty of Baylor University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Approved by the Dissertation Committee Brian E. Raines, D.Phil., Chairperson Nathan Alleman, Ph.D. Will Brian, D.Phil. Markus Hunziker, Ph.D. David Ryden, Ph.D. Accepted by the Graduate School August 2016 J. Larry Lyon, Ph.D., Dean Page bearing signatures is kept on file in the Graduate School. Copyright c 2016 by Reeve Hunter All rights reserved TABLE OF CONTENTS LIST OF FIGURES vi ACKNOWLEDGMENTS vii DEDICATION viii 1 Introduction 1 2 Preliminaries 4 2.1 Dynamical Systems . .4 2.2 Notions of Chaos . .5 2.3 The Specification Property . 13 2.4 Symbolic Dynamics and Shift Spaces . 20 2.5 Zd-Actions and Multidimensional Shift Spaces . 22 d 3 Chaos in Subshifts of ΣZ Via the Weak Specification Property 28 3.1 General !-Limit Sets for Zd-Actions . 28 d 3.2 Minimal Sets in Subshifts of ΣZ ..................... 29 3.3 !-Chaos . 37 3.4 Directional Limit Sets . 41 3.5 Directional Limit Chaos . 43 3.6 Distributional Chaos . 44 3.7 Some Implications . 51 iv 4 Specification and !-Chaos in a General Compact Metric Space 53 4.1 Some Definitions and Constructions . 54 4.2 Constructing a Set Exhibiting !-Chaos . 58 5 Future Work 63 5.1 More with !-Chaos . 63 5.2 Relations Between Various Types of Chaos for Zd-Actions . 64 5.3 Tiling Spaces . 64 5.4 Maps on Dendrites . 66 BIBLIOGRAPHY 70 v LIST OF FIGURES 2.1 Tracing orbits with the specification property . 16 2 2.2 An example of a point in ΣZ ...................... 25 d 2.3 The shift action for ΣZ ......................... 25 3.1 Block gluing . 31 3.2 Gluing multiple patches together . 33 d 3.3 “Exploded” point from ΣZ ........................ 34 2 3.4 !-chaos: Building shells in ΣZ ..................... 39 3 3.5 !-chaos: Building shells in ΣZ ..................... 40 3.6 Directional limit sets . 42 3.7 Directional limit sets for multiple directions . 43 2 3 3.8 Uniform DC1: Building shells in ΣZ and ΣZ ............. 47 d 3.9 Uniform DC1: Stripped shells in ΣZ .................. 48 4.1 !-chaos: Defining the sets U0 and U1 .................. 55 4.2 !-chaos: Using the specification property to construct points . 57 4.3 !-chaos: Finding limit points with controlled orbits . 59 5.1 An example of a Penrose tiling . 65 5.2 Graphs, trees, and dendrites . 67 5.3 A dendritic Julia set . 68 vi ACKNOWLEDGMENTS I would like to thank Brian Raines for giving me a wonderful introduction to Topology, an excellent example of understated mastery in the classroom, and the gift of agreeing to be my advisor. I appreciate your continual encouragement and steadfastness throughout my time here. Thanks for your valuable insights, your endless patience, and your genuine concern for me—both academically and personally. To my colleagues at Baylor: your varied perspectives helped me to break out of the narrow view I often found myself developing in both teaching and research. In particular, I would like to say thank you to Nathan and Tim for your help and companionship over the past five years. When things seemed especially bleak in the dark halls of Sid Richardson, we had Anamanaguchi’s “Meow” to thank for lifting our spirits. Come to think of it, any time was a good time for “Meow.” Thank you to my family for your support during my time in grad school. To Mom, Dad, Zellyn, Bevin, Trent, Leah, Delvyn, Charity, and Cheryl: though the times we have to spend together are too few, the many visits, phone calls, and conversations have been a precious, constant presence in my life. Finally, I must thank my wife, Jessie, for agreeing together with me to venture on this journey. Thank you for all your support, both practical and otherwise, your willingness to work towards a common goal when things seemed insurmountable, and most of all for your friendship. vii To Chuzzlewit viii CHAPTER ONE Introduction Topological dynamics, along with the related fields of ergodic theory and dif- ferentiable dynamics, has its roots in celestial mechanics. In 1609, Johannes Kepler, using data from the Danish astronomer Tycho Brahe, offered a fairly accurate ge- ometric description of the movement of planets in our solar system, [59]. In 1687, Isaac Newton presented general, mathematically rigorous principles describing the interaction of any two celestial bodies, [78]. Generalizations of Newton’s results to more than two bodies proved difficult to pin down. Modern topological dynamics, ergodic theory, and differentiable dynamics can all be traced back to Henri Poincaré’s work on the three body problem, the study of how three celestial bodies interact under the laws of physics. For a comprehensive history of Poincaré’s exploration of this problem, see [15]. Rather than focusing on the general structure of solutions to differential equa- tions based on their initial value, Poincaré chose instead to consider the big picture behavior and interactions of all possible individual solutions. In today’s terminol- ogy, he was considering the orbit structure of a system rather than focusing on a single orbit. The Poincaré Recurrence Theorem is the first major theorem in dy- namical systems that sprung from this view. For some interesting historical notes and references on this theorem, see [18]. Poincaré’s work had a profound influence on George Birkhoff who, in the early 1900’s, continued to apply this new approach while emphasizing the importance of discrete dynamical systems in understanding continuous differential equations, [20]. A nice explanation of the differences between 1 the traditional differential equations approach and the dynamical systems approach is outlined by Akin in his survey of Topological Dynamics, [3]. Gustav Arnold Hedlund continued in the vein of Birkhoff’s work bringing this new view of dynamical systems to a wider audience. Seeking to unify the discrete and continuous time views of dynamical systems, Hedlund with his student Walter Gottschalk addressed both views along with the general approach of topological group actions acting on space. In the mid 1950’s, these ideas and others were presented in the first work dedicated to the topological view of dynamics, [51]. As well as introducing the term “topological dynamics” to distinguish the study from the analytical perspective, Hedlund was also influential in the subfield of symbolic dynamical systems. With an interest in the chaotic properties of systems, one goal of this disserta- tion is to explore two of the notions mentioned in the previous paragraph: symbolic dynamics and group actions. Chaos theory can be traced back to Poincaré and the idea of sensitive dependence on initial conditions (more on this in Chapter Two). We focus on two notions formulated in the early 1990’s. In particular, we address d distributional chaos and !-chaos in the context of subshifts of ΣZ with a weak form of the specification property and !-chaos in the context of general compact metric spaces with the specification property. In Chapter Three, we address Zd-subshifts. d Theorem 3.8. Assume X ΣZ is a nondegenerate subshift with the weak specifi- ⊂ cation property. Then σ has !-chaos. d Theorem 3.15. If X ΣZ is a nondegenerate subshift with the weak specification ⊂ property then (X; σ) is uniformly distributionally chaotic of type 1. 2 Theorem 3.8 is further broadened to include some directional variants of !- chaos. In Chapter Four we prove a result for general compact metric spaces. Theorem 4.6. Assume that X is a compact metric space with f : X X a surjective ! map with the specification property. Assume s X is a fixed point such that there 2 is η > 0; λ > 1 such that if 0 < d(s; y) < η, then d(f(s); f(y)) λd(s; y). Then f is ≥ !-chaotic. In Chapter Two, we give some historical context for the problems presented in this dissertation as well as all necessary preliminary definitions and theorems. In Chapter Three, we introduce generalized !-limit sets and prove that the specification d property implies !-chaos for any ΣZ -subshifts. We show further that the same assumption implies uniform distributional chaos of type 1. It is known that an expansive dynamical system with the specification property has !-chaos. In Chapter Four, we show that specification along with the weakened assumption of uniform expansion at a fixed point is enough to imply !-chaos. Finally, we present some areas of further interest and ideas for future work in Chapter Five. 3 CHAPTER TWO Preliminaries 2.1 Dynamical Systems Throughout this dissertation, unless otherwise stated, X will be a compact metric space f : X X a continuous map.