Topics from One-Dimensional Dynamics

✍ Scribed by Karen M. Brucks, Henk Bruin

• Publisher: Cambridge University Press
• Year: 2012
• Tongue: English
• Leaves: 313
• Series: London Mathematical Society Student Texts, 62
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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