Topics from One-Dimensional Dynamics

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