Topics from one-dimensional dynamics / Karen M. Brucks, Henk Bruin.

Format:

Book

Author/Creator:

Brucks, Karen M. (Karen Marie), 1957-

Contributor:

Bruin, Henk, 1966-

Series:

London Mathematical Society student texts ; 62.

London Mathematical Society student texts ; 62

Language:

English

Subjects (All):

Differentiable dynamical systems.

Physical Description:

xiv, 297 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge, UK ; New York : Cambridge University Press, 2004.

Summary:

An introduction to one-dimensional dynamics for graduate students with novel connections to other areas of mathematics.

Contents:

1 Topological Roots 1

1.1 Basics from Topology 1

1.2 Middle Third Cantor Set 9

2 Measure Theoretic Roots 12

2.1 Basics of Lebesgue Measure on R 12

2.2 A Nonmeasurable Set 15

2.3 Lebesgue Measure of Cantor Sets 16

2.3.1 The Middle Third Cantor Set 16

2.3.2 Other Cantor Sets 17

2.4 Sets of Lebesgue Measure Zero 18

3 Beginning Symbolic and Topological Dynamics 19

3.1 Periodic Behavior 20

3.2 Nonwandering and [omega]-Limit Sets 24

3.3 Topological Conjugacy 33

3.4 Transitive Behavior 35

3.5 Recurrence 42

3.6 Shift Spaces 46

4 Beginning Measurable Dynamics 52

4.2 Measurable Maps on I 53

4.3 Poincare Recurrence 56

4.4 Ergodicity 58

4.4.1 Integration of Measurable Functions 60

4.4.2 Averaging Measurable Functions Along Orbits 62

4.4.3 A Connection to Topological Dynamics 65

5 A First Example: The 2[infinity] Map 66

5.1 Logistic Family 66

5.2 A Bit of Combinatorics 68

5.3 Construction of the Cantor Set [omega](c, g) 68

5.4 Cantor Set and Adding Machines 70

5.5 A Toeplitz Sequence 73

6 Kneading Maps 74

6.1 Hofbauer Towers and Kneading Maps 74

6.2 First Uses of Kneading Maps 80

6.3 Shadowing 85

6.4 Examples of Kneading Maps 87

7 Some Number Theory 92

7.1 Farey Tree 92

7.2 Continued Fractions 96

7.3 Continued Fractions and the Farey Tree 98

8 Circle Maps 101

8.1 Circle Homeomorphisms 101

8.2 Degree One Circle Maps 105

8.3 Irrational Rotations and Return Maps 111

8.4 Cantor Thread 114

9 Topological Entropy 117

9.1 Basic Properties of Topological Entropy 118

9.2 Entropy of Subshifts 123

9.3 Lapnumbers and Markov Extensions 129

9.4 Lapnumbers and Entropy 136

9.5 Semiconjugacy to a Piecewise Linear Map 140

9.6 The Monotonicity Problem 144

10 Symmetric Tent Maps 147

10.1 Preliminary Combinatorics 148

10.2 [omega]-Limit Sets 153

10.3 Phase Portrait 157

10.4 Measure Results 165

10.5 Slow Recurrence and the CE Condition 168

10.6 Attractors 172

10.7 Combinatorics and Renormalization 174

11 Unimodal Maps and Rigid Rotations 178

11.1 Adding Machines in Unimodal Maps 178

11.2 Rigid Rotations in Unimodal Maps - I 184

11.3 Rigid Rotations in Unimodal Maps - II 186

12 [beta]-Transformations, Unimodal Maps, and Circle Maps 193

12.1 [beta]-Transformations and [beta]-Expansions 193

12.2 Flip-Half-of-the-Graph Trick 195

12.3 A Relation Between Unimodal Maps and Circle Maps 197

12.4 Comparing [beta]-Transformations and Tent Maps 203

12.5 Ledrappier's Example 208

12.6 Maps with Slope [less than sign] 2 211

13 Homeomorphic Restrictions in the Unimodal Setting 216

13.1 First Observations 218

13.2 A 2[infinity] Trapezoidal Map 220

13.3 The Adding Machine ([Omega], P) 225

13.4 The Case Q(k) to [infinity] 238

14 Complex Quadratic Dynamics 250

14.1 Julia Sets and External Rays 251

14.2 The Mandelbrot Set 262

14.3 Itineraries and Hubbard Trees 266.

Notes:

Includes bibliographical references (pages 279-291) and index.

ISBN:

0521838967

0521547660

OCLC:

53940500

Online:

Publisher description