Book description
One-dimensional dynamics owns many deep results and avenues of active mathematical research. Numerous inroads to this research exist for the advanced undergraduate or beginning graduate student. This book provides glimpses into one-dimensional dynamics with the hope that the results presented illuminate the beauty and excitement of the field. Much of this material is covered nowhere else in textbook format, some are mini new research topics in themselves, and novel connections are drawn with other research areas both inside and outside the text. The material presented here is not meant to be approached in a linear fashion. Readers are encouraged to pick and choose favourite topics. Anyone with an interest in dynamics, novice or expert alike, will find much of interest within.
Reviews
‘… particularly useful for students/beginners in the field. Due to an extensive bibliography, it will also serve as a very good reference book.‘
Source: European Mathematical Society Newsletter
'This book is intended as a text for an advanced undergraduate or beginning graduate students. As well as providing a brief account on the fundamental concepts of analysis and dynamical systems (Chapters 1-4 and 7-8), and a thorough explanation of topological entropy for piece-wise monotone interval maps (Chapter 9, sometimes with original proofs), the book contains substantial parts on unimodal interval maps and one chapter on complex quadratic polynomials. The quality of this exposition is very good: the material is organized so that all proofs are of easily manageable length; motivational comments are helpful, and there are plenty of illustrative examples and exercises.'
Source: Zentralblatt MATH
Author(s): Karen M. Brucks, Henk Bruin
Series: London Mathematical Society Student Texts (62)
Publisher: Cambridge University Press
Year: 2012
Language: English
Pages: 314
Tags: Dynamical systems, One-dimensional dynamics
Contents
List of Figures
Preface
1 Topological Roots
1.1 Basics from Topology
1.2 Middle Third Cantor Set
2 Measure Theoretic Roots 12
2.1 Basics of Lebesgue Measure on 1R
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
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
4 Beginning Measurable Dynamics
4.1 Preliminaries
4.2 Measurable Maps on I
4.3 Poincare Recurrence
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
5 A First Example: The 2°°Map
5.1 Logistic Family
5.2 A Bit of Combinatorics
5.3 Construction of the Cantor Set w(c,g)
5.4 Cantor Set and Adding Machines
5.5 A Toeplitz Sequence
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
7 Some Number Theory
7.1 Farey Tree
7.2 Continued Fractions
7.3 Continued Fractions and the Farey Tree
8 Circle Maps
8.1 Circle Homeomorphisms
8.2 Degree One Circle Maps
8.3 Irrational Rotations and Return Maps
8.4 Cantor Thread
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
10 Symmetric Tent Maps
10.1 Preliminary Combinatorics
10.2 w—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
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
12 fi—Transformations, Unimodal Maps, and Circle Maps
12.1 fi—Transformations and fi—Expansions
12.2 Flip—Half—of—the-Graph Trick
12.3 A Relation Between Unimodal Maps and Circle Maps
12.4 Comparing fi-Transformations and Tent Maps
12.5 Ledrappier's Example
12.6 Maps with Slope < 2
13 Homeomorphic Restrictions in the Unimodal Setting
13.1 First Observations
13.2 A 2°° Trapezoidal Map
13.3 The Adding Machine (Q,P)
13.4 The Case Q(k)--> + oo
14 Complex Quadratic Dynamics
14.1 Julia Sets and External Rays
14.2 The Mandelbrot Set
14.3 Itineraries and Hubbard Trees
Bibliography
Index
