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Chaotic Billiards

Publisher: 
American Mathematical Society
Number of Pages: 
316
Price: 
85.00
ISBN: 
0821840967
Date Received: 
Monday, July 31, 2006
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Nikolai Chernov and Roberto Markarian
Series: 
Mathematical Surveys and Monographs 127
Publication Date: 
2006
Format: 
Hardcover
Category: 
Monograph

Preface ix

Symbols and notation xi

Chapter 1. Simple examples 1

1.1. Billiard in a circle 1

1.2. Billiard in a square 5

1.3. A simple mechanical model 9

1.4. Billiard in an ellipse 11

1.5. A chaotic billiard: pinball machine 15

Chapter 2. Basic constructions 19

2.1. Billiard tables 19

2.2. Unbounded billiard tables 22

2.3. Billiard flow 23

2.4. Accumulation of collision times 24

2.5. Phase space for the flow 26

2.6. Coordinate representation of the flow 27

2.7. Smoothness of the flow 29

2.8. Continuous extension of the flow 30

2.9. Collision map 31

2.10. Coordinates for the map and its singularities 32

2.11. Derivative of the map 33

2.12. Invariant measure of the map 35

2.13. Mean free path 37

2.14. Involution 38

Chapter 3. Lyapunov exponents and hyperbolicity 41

3.1. Lyapunov exponents: general facts 41

3.2. Lyapunov exponents for the map 43

3.3. Lyapunov exponents for the flow 45

3.4. Hyperbolicity as the origin of chaos 48

3.5. Hyperbolicity and numerical experiments 50

3.6. Jacobi coordinates 51

3.7. Tangent lines and wave fronts 52

3.8. Billiard-related continued fractions 55

3.9. Jacobian for tangent lines 57

3.10. Tangent lines in the collision space 58

3.11. Stable and unstable lines 59

3.12. Entropy 60

v

vi CONTENTS

3.13. Proving hyperbolicity: cone techniques 62

Chapter 4. Dispersing billiards 67

4.1. Classification and examples 67

4.2. Another mechanical model 69

4.3. Dispersing wave fronts 71

4.4. Hyperbolicity 73

4.5. Stable and unstable curves 75

4.6. Proof of Proposition 4.29 77

4.7. More continued fractions 83

4.8. Singularities (local analysis) 86

4.9. Singularities (global analysis) 88

4.10. Singularities for type B billiard tables 91

4.11. Stable and unstable manifolds 93

4.12. Size of unstable manifolds 95

4.13. Additional facts about unstable manifolds 97

4.14. Extension to type B billiard tables 99

Chapter 5. Dynamics of unstable manifolds 103

5.1. Measurable partition into unstable manifolds 103

5.2. u-SRB densities 104

5.3. Distortion control and homogeneity strips 107

5.4. Homogeneous unstable manifolds 109

5.5. Size of H-manifolds 111

5.6. Distortion bounds 113

5.7. Holonomy map 118

5.8. Absolute continuity 120

5.9. Two growth lemmas 124

5.10. Proofs of two growth lemmas 127

5.11. Third growth lemma 132

5.12. Size of H-manifolds (a local estimate) 136

5.13. Fundamental theorem 137

Chapter 6. Ergodic properties 141

6.1. History 141

6.2. Hopf’s method: heuristics 141

6.3. Hopf’s method: preliminaries 143

6.4. Hopf’s method: main construction 144

6.5. Local ergodicity 147

6.6. Global ergodicity 151

6.7. Mixing properties 152

6.8. Ergodicity and invariant manifolds for billiard flows 154

6.9. Mixing properties of the flow and 4-loops 156

6.10. Using 4-loops to prove K-mixing 158

6.11. Mixing properties for dispersing billiard flows 160

Chapter 7. Statistical properties 163

7.1. Introduction 163

7.2. Definitions 163

7.3. Historic overview 167

CONTENTS vii

7.4. Standard pairs and families 169

7.5. Coupling lemma 172

7.6. Equidistribution property 175

7.7. Exponential decay of correlations 176

7.8. Central Limit Theorem 179

7.9. Other limit theorems 184

7.10. Statistics of collisions and diffusion 186

7.11. Solid rectangles and Cantor rectangles 190

7.12. A ‘magnet’ rectangle 193

7.13. Gaps, recovery, and stopping 197

7.14. Construction of coupling map 200

7.15. Exponential tail bound 205

Chapter 8. Bunimovich billiards 207

8.1. Introduction 207

8.2. Defocusing mechanism 207

8.3. Bunimovich tables 209

8.4. Hyperbolicity 210

8.5. Unstable wave fronts and continued fractions 214

8.6. Some more continued fractions 216

8.7. Reduction of nonessential collisions 220

8.8. Stadia 223

8.9. Uniform hyperbolicity 227

8.10. Stable and unstable curves 230

8.11. Construction of stable and unstable manifolds 232

8.12. u-SRB densities and distortion bounds 235

8.13. Absolute continuity 238

8.14. Growth lemmas 242

8.15. Ergodicity and statistical properties 248

Chapter 9. General focusing chaotic billiards 251

9.1. Hyperbolicity via cone techniques 252

9.2. Hyperbolicity via quadratic forms 254

9.3. Quadratic forms in billiards 255

9.4. Construction of hyperbolic billiards 257

9.5. Absolutely focusing arcs 260

9.6. Cone fields for absolutely focusing arcs 263

9.7. Continued fractions 265

9.8. Singularities 266

9.9. Application of Pesin and Katok-Strelcyn theory 270

9.10. Invariant manifolds and absolute continuity 272

9.11. Ergodicity via ‘regular coverings’ 274

Afterword 279

Appendix A. Measure theory 281

Appendix B. Probability theory 291

Appendix C. Ergodic theory 299

Publish Book: 
Modify Date: 
Monday, July 31, 2006

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