Chaos and chance : an introduction to stochastic apects of dynamics / Arno Berger.

Format:

Book

Author/Creator:

Berger, Arno, 1968-

Series:

De Gruyter textbook

Language:

English

Subjects (All):

Differentiable dynamical systems.

Chaotic behavior in systems.

Physical Description:

x, 245 pages : illustrations ; 24 cm.

Place of Publication:

Berlin ; New York : Walter de Gruyter, 2001.

Contents:

1.1 Long-time behaviour of mechanical systems 1

1.2 Iteration of maps 9

1.3 Elementary stochastic processes 14

2 Basic aspects of discrete dynamical systems 20

2.1 Hyperbolicity and bifurcations 20

2.2 How may simple systems become complicated? 27

2.3 Facing deterministic chaos 31

2.3.1 Symbolic dynamical systems 31

2.3.2 The emergence of chaos 37

2.3.3 Newton's method for polynomials: a case study 43

2.4 Circle maps, rotation numbers, and minimality 48

2.5 Glimpses of billiards 54

2.6 Horseshoes, attractors, and natural extensions 61

2.7 Toral maps and shadowing 68

3 Ergodic theory I. Foundations 79

3.1 The statistical point of view 79

3.2 Invariant and ergodic measures 84

3.3 Ergodic theorems 94

3.4 Aspects of mixing 109

3.4.1 Mixing properties 110

3.4.2 The concept of entropy 112

4 Ergodic theory II. Applications 126

4.1 The Frobenius-Perron operator 126

4.2 Asymptotic behaviour of densities 135

4.3 Piecewise expanding Markov maps 145

4.4 A short look at Markov chains 156

4.4.1 Class structure, absorption probabilities, and hitting times 161

4.4.2 Recurrence and transience: dynamical classification of states 164

4.4.3 The long-time behaviour of Markov chains 167

5 The dynamical evolution of measures 179

5.1 Basic examples and concepts 179

5.2 Asymptotic stability 187

5.3 Back to geometry: fractal sets and measures 191

5.4 Three final examples 205

5.4.1 Searching for non-normal numbers 205

5.4.2 The fractal nature of Brownian paths 207

5.4.3 Patterns of congruence in the Pascal triangle 214

A The toolbox 221

A.1 A survey of notations 221

A.2 Basic facts from measure theory 223

A.3 Integration theory 226

A.4 Conditional expectations 230.

Notes:

Includes bibliographical references (pages [237]-240) and index.

ISBN:

3110169908

OCLC:

47177018