**Proceedings of Symposia in Pure Mathematics**

Volume: 89;
2015;
339 pp;
Hardcover

MSC: Primary 37; 60;

Print ISBN: 978-1-4704-1112-1

Product Code: PSPUM/89

List Price: $120.00

Individual Member Price: $96.00

**Electronic ISBN: 978-1-4704-2266-0
Product Code: PSPUM/89.E**

List Price: $120.00

Individual Member Price: $96.00

# Hyperbolic Dynamics, Fluctuations and Large Deviations

Share this page *Edited by *
*D. Dolgopyat; Y. Pesin; M. Pollicott; L. Stoyanov*

This volume contains the proceedings of the semester-long special program on Hyperbolic Dynamics, Large Deviations and Fluctuations, which was held from January–June 2013, at the Centre Interfacultaire Bernoulli, École Polytechnique Fédérale de Lausanne, Switzerland.

The broad theme of the program was the long-term behavior of dynamical systems and their statistical behavior. During the last 50 years, the statistical properties of dynamical systems of many different types have been the subject of extensive study in statistical mechanics and thermodynamics, ergodic and probability theories, and some areas of mathematical physics. The results of this study have had a profound effect on many different areas in mathematics, physics, engineering and biology.

The papers in this volume cover topics in large deviations and thermodynamics formalism and limit theorems for dynamic systems.

The material presented is primarily directed at researchers and graduate students in the very broad area of dynamical systems and ergodic theory, but will also be of interest to researchers in related areas such as statistical physics, spectral theory and some aspects of number theory and geometry.

#### Table of Contents

# Table of Contents

## Hyperbolic Dynamics, Fluctuations and Large Deviations

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Introduction 110
- Large Deviations and Thermodynamical Formalism 716
- The almost Borel structure of diffeomorphisms with some hyperbolicity 918
- Lectures on large deviations in probability and dynamical systems 4554
- Thermodynamic formalism for countable Markov shifts 8190
- 1. What is “thermodynamic formalism”? 8190
- 2. Topological Markov shifts (TMS) 8897
- 3. Conformal measures and their ergodic properties 9099
- 4. Ruelle’s operator, thermodynamic limits, and modes of recurrence 93102
- 5. Pressure, equilibrium measures, and Gibbs measures in the sense of Bowen 97106
- 6. Strong positive recurrence and spectral gap 100109
- 7. Absence of spectral gap and critical phenomena 105114
- 8. Application to surface diffeomorphisms 108117
- References 112121

- Limit Theorems for Dynamical Systems 119128
- Limit theorems for horocycle flows 121130
- Limit theorems in dynamical systems using the spectral method 161170
- Kinetic limits of dynamical systems 195204
- 1. Motivation 195204
- 2. The Lorentz gas 196205
- 3. Mean free path length in the Lorentz gas 198207
- 4. The Boltzmann-Grad limit of the Lorentz gas 199208
- 5. Kicked Hamiltonians 203212
- 6. Geometric representation 204213
- 7. Mean collision time in kicked Hamiltonians 205214
- 8. The Boltzmann-Grad limit of kicked Hamiltonians 206215
- 9. Renormalisation of the transition kernel for the Lorentz gas 206215
- 10. Renormalisation of the transition kernel for kicked Hamiltonians 209218
- 11. Poisson process 209218
- 12. Point processes and homogeneous spaces 210219
- 13. The space of lattices 210219
- 14. The transition kernel for lattices 212221
- 15. The distribution of free path lengths 214223
- 16. Cut-and-project sets 214223
- 17. Spaces of cut-and-project sets 215224
- 18. Equidistribution 216225
- 19. Examples of cut-and-project sets and their \SLdR-closures 217226
- 20. Conclusions 220229
- Acknowledgements 221230
- References 221230

- Additional Topics 225234
- Limit theorems for toral translations 227236
- 1. Introduction 228237
- 2. Ergodic sums of smooth functions with singularities 229238
- 3. Ergodic sums of characteristic functions. Discrepancies 233242
- 4. Poisson regime 237246
- 5. Poisson processes 237246
- 6. Uniform distribution on the space of lattices 239248
- 7. Ideas of the Proofs 241250
- 8. Shrinking targets 248257
- 9. Skew products. Random walks. 252261
- 10. Special flows. 259268
- 11. Higher dimensional actions 268277
- References 272281

- Spectral gap properties and limit theorems for some random walks and dynamical systems 279288
- Introduction 279288
- 1. Spectral gap properties of Markov operators and the local limit theorem. 280289
- 2. Law of large numbers and spectral gap properties for products of random matrices. 284293
- 3. Local limit theorems for some transfer operators 298307
- 4. Extreme value theory for affine random walks. 303312
- References 308317

- The martingale approach after Varadhan and Dolgopyat 311320
- 1. Introduction 312321
- 2. Preliminaries and results 314323
- 3. Standard Pairs 317326
- 4. Conditioning 319328
- 5. Averaging (the Law of Large Numbers) 321330
- 6. A recap of what we have done so far 324333
- 7. Fluctuations (the Central Limit Theorem) 324333
- Appendix A. Geometry 334343
- Appendix B. Shadowing 335344
- Appendix C. Martingales, operators and Itō’s calculus 336345
- References 338347

- Back Cover Back Cover1354

#### Readership

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