**Contemporary Mathematics**

Volume: 469;
2008;
340 pp;
Softcover

MSC: Primary 37; 60; 70;

**Print ISBN: 978-0-8218-4286-7
Product Code: CONM/469**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-0-8218-8148-4
Product Code: CONM/469.E**

List Price: $105.00

AMS Member Price: $84.00

MAA Member Price: $94.50

# Geometric and Probabilistic Structures in Dynamics

Share this page *Edited by *
*Keith Burns; Dmitry Dolgopyat; Yakov Pesin*

This book presents a collection of articles that cover areas of mathematics
related to dynamical systems. The authors are well-known experts who use
geometric and probabilistic methods to study interesting problems in
the theory of dynamical systems and its applications. Some of the articles
are surveys while others are original contributions. The topics covered
include: Riemannian geometry, models in mathematical physics and
mathematical biology, symbolic dynamics, random and stochastic dynamics.

This book can be used by graduate students and researchers in dynamical
systems and its applications.

#### Readership

Graduate students and research mathematicians interested in dynamical systems and geometry.

# Table of Contents

## Geometric and Probabilistic Structures in Dynamics

- Contents vii8 free
- Preface ix10 free
- Nonconvergence examples in averaging 118 free
- Periodic rank one geodesics in Hadamard spaces 1936
- Dynamical determinants and spectrum for hyperbolic diffeomorphisms 2946
- Open problems in symbolic dynamics 6986
- 1. Introduction 7087
- 2. Definitions 7188
- 3. Classification of SFTs and the Little Shift Equivalence Conjecture 7390
- 4. Range of the dimension representation 7491
- 5. Positive Rational Shift Equivalence Conjecture 7592
- 6. The Generalized Spectral Conjecture 7693
- 7. The Equal Entropy Factors Conjecture for SFTs 7794
- 8. The factors problem for sofic shifts 7794
- 9. The Good Finitary Conjecture for Markov shifts 7895
- 10. What can be a beta function? 8097
- 11. Structure of expansive subdynamics 8097
- 12. Can nonSFT expansive maps commute with irreducible SFTs? 8299
- 13. The Commuting Powers Conjecture for SFTs 8299
- 14. Jointly invariant measures 83100
- 15. Zd SFTs 84101
- 16. Stable limit sets of cellular automata 85102
- 17. Algebraic Zd SFTs 86103
- 18. Finitary codes and Markov random fields 86103
- 19. Decidability problems 88105
- 20. Onesided SFTs: the embedding problem 89106
- 21. Onesided sofic shifts: the classification problem 89106
- 22. The Virtual FOG Conjecture for automorphisms of a mixing SFT 91108
- 23. Topological orbit equivalence 92109
- 24. Symbolic extension entropy and entropy structure 93110
- 25. Cellular automata and periodic points 94111
- 26. Cellular automata on big groups 95112
- 27. Entropy conjugacy and countable state Markov chains 95112
- 28. Beta shifts: Salem numbers and intrinsic ergodicity 96113
- 29. Adler's Renewal Question 96113
- 30. The Road Coloring Problem 96113
- 31. Parry's Finiteness Question for skew products 97114
- 32. Classification of sofic shifts 97114
- 33. Classification and flow equivalence of general shifts 99116
- 34. The Pisot Conjecture 101118
- 35. Nivat's Conjecture 102119
- Appendices 102119
- Appendix A. Commuting SFTs and periodic points 102119
- Appendix B. Commuting SFTs from commuting matrices, following Nasu 104121
- Appendix C. LR Textile Systems 105122
- Appendix D. Commuting SFTs from matrices commuting on dimension 107124
- Appendix E. Primitive matrices SE but not SSE over Z+ [1/p] 109126
- Appendix F. Examples for the onesided sofic classification problem 110127
- Bibliography 111128

- New examples of topologically equivalent S-unimodal maps with different metric properties 119136
- Appendix. Quasiconformal deformation of multipliers 135152
- A remark on the group of PL-homeomorphisms in dimension one 141158
- Fermi acceleration 149166
- Riemannian 2-step nilmanifolds with prescribed Ricci tensor 167184
- On the Greenfield-Wallach and Katok conjectures in dimension three 197214
- Metastability and Stochastic Resonance for Multiscale Systems 215232
- Mendelian Dynamics and Sturtevant's Paradigm 227244
- On the estimation of topological entropy on surfaces 243260
- 1. Introduction 243260
- 2. Definition and properties of Topological Entropy 244261
- 3. Trellises and Entropy 246263
- 4. Computing stable and unstable manifolds 250267
- 5. Computation of the stable and unstable manifolds in Henon Maps 251268
- 6. Rigorous Topological Arguments in the Plane with Taylor Models 257274
- 7. An alternate numerical description of the rectangles Ri and their properties 265282
- References 269286

- Dynamical Systems and Differential Forms. Low Dimensional Hamiltonian Systems 271288
- Periodic orbits and holonomy for hyperbolic flows 289306
- On existence and uniqueness of weak foliations in dimension 3 303320
- A Limit Theorem for Birkhoff sums of non-integrable functions over rotations 317334