**Mathematical Surveys and Monographs**

Volume: 127;
2006;
316 pp;
Hardcover

MSC: Primary 37;
Secondary 82

Print ISBN: 978-0-8218-4096-2

Product Code: SURV/127

List Price: $93.00

Individual Member Price: $74.40

**Electronic ISBN: 978-1-4704-1354-5
Product Code: SURV/127.E**

List Price: $93.00

Individual Member Price: $74.40

#### Supplemental Materials

# Chaotic Billiards

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*Nikolai Chernov; Roberto Markarian*

This book covers one of the most exciting but most difficult topics in the modern theory of dynamical systems: chaotic billiards. In physics, billiard models describe various mechanical processes, molecular dynamics, and optical phenomena.

The theory of chaotic billiards has made remarkable progress in the past thirty-five years, but it remains notoriously difficult for the beginner, with main results scattered in hardly accessible research articles. This is the first and so far only book that covers all the fundamental facts about chaotic billiards in a complete and systematic manner. The book contains all the necessary definitions, full proofs of all the main theorems, and many examples and illustrations that help the reader to understand the material. Hundreds of carefully designed exercises allow the reader not only to become familiar with chaotic billiards but to master the subject.

The book addresses graduate students and young researchers in physics and mathematics. Prerequisites include standard graduate courses in measure theory, probability, Riemannian geometry, topology, and complex analysis. Some of this material is summarized in the appendices to the book.

#### Table of Contents

# Table of Contents

## Chaotic Billiards

- Contents v6 free
- Preface ix10 free
- Symbols and notation xi12 free
- Chapter 1. Simple examples 114 free
- Chapter 2. Basic constructions 1932
- 2.1. Billiard tables 1932
- 2.2. Unbounded billiard tables 2235
- 2.3. Billiard flow 2336
- 2.4. Accumulation of collision times 2437
- 2.5. Phase space for the flow 2639
- 2.6. Coordinate representation of the flow 2740
- 2.7. Smoothness of the flow 2942
- 2.8. Continuous extension of the flow 3043
- 2.9. Collision map 3144
- 2.10. Coordinates for the map and its singularities 3245
- 2.11. Derivative of the map 3346
- 2.12. Invariant measure of the map 3548
- 2.13. Mean free path 3750
- 2.14. Involution 3851

- Chapter 3. Lyapunov exponents and hyperbohcity 4154
- 3.1. Lyapunov exponents: general facts 4154
- 3.2. Lyapunov exponents for the map 4356
- 3.3. Lyapunov exponents for the flow 4558
- 3.4. Hyperbohcity as the origin of chaos 4861
- 3.5. Hyperbohcity and numerical experiments 5063
- 3.6. Jacobi coordinates 5164
- 3.7. Tangent lines and wave fronts 5265
- 3.8. Billiard-related continued fractions 5568
- 3.9. Jacobian for tangent lines 5770
- 3.10. Tangent lines in the collision space 5871
- 3.11. Stable and unstable lines 5972
- 3.12. Entropy 6073
- 3.13. Proving hyperbolicity: cone techniques 6275

- Chapter 4. Dispersing billiards 6780
- 4.1. Classification and examples 6780
- 4.2. Another mechanical model 6982
- 4.3. Dispersing wave fronts 7184
- 4.4. Hyperbolicity 7386
- 4.5. Stable and unstable curves 7588
- 4.6. Proof of Proposition 4.29 7790
- 4.7. More continued fractions 8396
- 4.8. Singularities (local analysis) 8699
- 4.9. Singularities (global analysis) 88101
- 4.10. Singularities for type B billiard tables 91104
- 4.11. Stable and unstable manifolds 93106
- 4.12. Size of unstable manifolds 95108
- 4.13. Additional facts about unstable manifolds 97110
- 4.14. Extension to type B billiard tables 99112

- Chapter 5. Dynamics of unstable manifolds 103116
- 5.1. Measurable partition into unstable manifolds 103116
- 5.2. u-SRB densities 104117
- 5.3. Distortion control and homogeneity strips 107120
- 5.4. Homogeneous unstable manifolds 109122
- 5.5. Size of H-manifolds 111124
- 5.6. Distortion bounds 113126
- 5.7. Holonomy map 118131
- 5.8. Absolute continuity 120133
- 5.9. Two growth lemmas 124137
- 5.10. Proofs of two growth lemmas 127140
- 5.11. Third growth lemma 132145
- 5.12. Size of H-manifolds (a local estimate) 136149
- 5.13. Fundamental theorem 137150

- Chapter 6. Ergodic properties 141154
- 6.1. History 141154
- 6.2. Hopf's method: heuristics 141154
- 6.3. Hopf's method: preliminaries 143156
- 6.4. Hopf's method: main construction 144157
- 6.5. Local ergodicity 147160
- 6.6. Global ergodicity 151164
- 6.7. Mixing properties 152165
- 6.8. Ergodicity and invariant manifolds for billiard flows 154167
- 6.9. Mixing properties of the flow and 4-loops 156169
- 6.10. Using 4-loops to prove K-mixing 158171
- 6.11. Mixing properties for dispersing billiard flows 160173

- Chapter 7. Statistical properties 163176
- 7.1. Introduction 163176
- 7.2. Definitions 163176
- 7.3. Historic overview 167180
- 7.4. Standard pairs and families 169182
- 7.5. Coupling lemma 172185
- 7.6. Equidistribution property 175188
- 7.7. Exponential decay of correlations 176189
- 7.8. Central Limit Theorem 179192
- 7.9. Other limit theorems 184197
- 7.10. Statistics of collisions and diffusion 186199
- 7.11. Solid rectangles and Cantor rectangles 190203
- 7.12. A 'magnet' rectangle 193206
- 7.13. Gaps, recovery, and stopping 197210
- 7.14. Construction of coupling map 200213
- 7.15. Exponential tail bound 205218

- Chapter 8. Bunimovich billiards 207220
- 8.1. Introduction 207220
- 8.2. Defocusing mechanism 207220
- 8.3. Bunimovich tables 209222
- 8.4. Hyperbolicity 210223
- 8.5. Unstable wave fronts and continued fractions 214227
- 8.6. Some more continued fractions 216229
- 8.7. Reduction of nonessential collisions 220233
- 8.8. Stadia 223236
- 8.9. Uniform hyperbolicity 227240
- 8.10. Stable and unstable curves 230243
- 8.11. Construction of stable and unstable manifolds 232245
- 8.12. u-SRB densities and distortion bounds 235248
- 8.13. Absolute continuity 238251
- 8.14. Growth lemmas 242255
- 8.15. Ergodicity and statistical properties 248261

- Chapter 9. General focusing chaotic billiards 251264
- 9.1. Hyperbolicity via cone techniques 252265
- 9.2. Hyperbolicity via quadratic forms 254267
- 9.3. Quadratic forms in billiards 255268
- 9.4. Construction of hyperbolic billiards 257270
- 9.5. Absolutely focusing arcs 260273
- 9.6. Cone fields for absolutely focusing arcs 263276
- 9.7. Continued fractions 265278
- 9.8. Singularities 266279
- 9.9. Application of Pesin and Katok-Strelcyn theory 270283
- 9.10. Invariant manifolds and absolute continuity 272285
- 9.11. Ergodicity via 'regular coverings' 274287

- Afterword 279292
- Appendix A. Measure theory 281294
- Appendix B. Probability theory 291304
- Appendix C. Ergodic theory 299312
- Bibliography 309322
- Index 315328 free

#### Readership

Graduate students and research mathematicians interested in mathematical physics, statistical mechanics, dynamical systems, and ergodic theory.

#### Reviews

Although there are many books covering general mathematical billiards there are no comprehensive introductory texts covering chaotic billiards. The book remedies this deficiency and presents the theory of chaotic billiards in a systematic way.

-- Zentralblatt Math

In contrast to many other works on billiards, this book does not hide any technicalities. It works out all the technical details, to the last epsilon. There are numerous exercises which are well chosen and useful. This way the reader can not only understand billiard theory in an active way, but can also develop the skills needed to address new questions. Thus it is useful not only graduate students, but also for senior mathematicans who would like to start working on billiards. I strongly recommend this book.

-- Mathematical Reviews