Topics from One-Dimensional Dynamics

Description

One-dimensional dynamics has generated many results, and avenues of active mathematical research with numerous inroads to this research remain to be pursued by the advanced undergraduate or beginning graduate student. While much of the material in this book is not covered elsewhere, some aspects present new research topics whose connections are drawn to other research areas from the text. Although the material presented is not meant to be approached in a linear fashion, anybody with an interest in dynamics will find many topics of interest. Cover Series Page Title Contents List of Figures Preface Chapter 1 Topological Roots 1.1 Basics from Topology 1. 2 The Middle Third Cantor Set Chapter 2 Measure Theoretic Roots 2.1 Basics of Lebesgue Measure on R 2.2 A Nonmeasurable Set 2.3 Lebesgue Measure of Cantor Sets 2.3.1 The Middle Third Cantor Set 2.3.2 Other Cantor Sets 2.4 Sets of Lebesgue Measure Zero Chapter 3 Beginning Symbolic and Topological Dynamics 3.1 Periodic Behavior 3.2 Nonwandering and w-Limit Sets 3.3 Topological Conjugacy 3.4 Transitive Behavior 3. 5 Recurrence 3.6 Shift Spaces Chapter 4 Beginning Measurable Dynamics 4.1 Preliminaries 4.2 Measurable Maps on I 4.4 Ergodicity 4.4.1 Integration of Measurable Functions 4.4.2 Averaging Measurable Functions. Along Orbits 4.4.3 A Connection to Topological Dynamics Chapter 5 A First Example: The 2∞ Map 5.1 Logistic Family 5.2 A Bit of Combinatorics 5.3 Construction of the Cantor Set ω(c, g) 5.4 Cantor Set and Adding Machines 5.5 A Toeplitz Sequence Chapter 6 Kneading Maps 6.1 Hofbauer Towers and Kneading Maps 6.2 First Uses of Kneading Maps 6.3 Shadowing 6.4 Examples of Kneading Maps Chapter 7 Some Number Theory 7.1 The Farey Tree 7.2 Continued Fractions 7.3 Continued Fractions and the Farey Tree Chapter 8 Circle Maps 8.1 Circle Homeomorphisms 8.2 Degree One Circle Maps 8.3 Irrational Rotations and Return Maps 8.4 Cantor Thread Chapter 9 Topological Entropy 9.1 Basic Properties of Topological Entropy 9.2 Entropy of Subshifts 9.3 Lapnumbers and Markov Extensions 9.4 Lapnumbers and Entropy 9.5 Semiconjugacy to a Piecewise Linear Map 9.6 The Monotonicity Problem Chapter 10 Symmetric Tent Maps 10.1 Preliminary Combinatorics 10.2 ω-Limit Sets 10.3 Phase Portrait 10.4 Measure Results 10.5 Slow Recurrence and the CE Condition 10.6 Attractors 10.7 Combinatorics and Renormalization Chapter 11 Unimodal Maps and Rigid Rotations 11.1 Adding Machines in Unimodal Maps 11.2 Rigid Rotations in Unimodal Maps – I 11.3 Rigid Rotations in Unimodal Maps – II Chapter 12 β-Transformations, Unimodal Maps, and Circle Maps 12.1 β-Transformations and β-Expansions 12.2 Flip-Half-of-the-Graph Trick 12.3 A Relation Between Unimodal Maps and Circle Maps 12.4 Comparing /3-Transformations and Tent Maps 12.5 Ledrappier's Example 12.6 Maps with Slope < 2 Chapter 13 Homeomorphic Restrictions the Unimodal Setting 13.1 First Observations 13.2 A 2∞ Trapezoidal Map 13.3 The Adding Machine (Ω, P) 13.4 The Case Q(k) → ∞ Chapter 14 Complex Quadratic Dynamics 14.1 Julia Sets and External Rays 14.2 The Mandelbrot Set 14.3 Itineraries and Hubbard Trees Bibliography Index