Dynamics, ergodic theory, and geometry

✍ Scribed by Boris Hasselblatt

• Publisher: Cambridge University Press
• Year: 2007
• Tongue: English
• Leaves: 335
• Series: Mathematical Sciences Research Institute Publications 54
• Category: Library

✦ Synopsis

This volume contains surveys and research articles by leading experts in several areas of dynamical systems that have recently experienced substantial progress. Some of the major surveys focus on symplectic geometry; smooth rigidity; hyperbolic, parabolic, and symbolic dynamics; and ergodic theory. Students and researchers in dynamical systems, geometry, and related areas will find this a fascinating look at the state of the art.

✦ Table of Contents

Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Dynamics, Ergodic Theory, and Geometry......Page 6
ISBN-13 9780511342851 ISBN-10 0511342853 ISBN-13 9780521875417 ISBN-10 0521875412......Page 7
Contents......Page 8
Foreword......Page 10
Quantitative symplectic geometry......Page 12
1. Symplectic geometry and its neighbors......Page 13
2. Examples of symplectic capacities......Page 16
3. General properties and relations between symplectic capacities......Page 23
4. Ellipsoids and polydiscs......Page 36
References......Page 50
1. Prologue......Page 56
2. A brief digression: some examples of groups and actions......Page 60
3. Prehistory......Page 65
4. History......Page 71
5. Recent developments......Page 83
6. Directions for future research and conjectures......Page 93
References......Page 100
1. Introduction......Page 110
2. Moyennabilité relative......Page 112
3. Le lemme de remplissage......Page 114
4. Dèmonstration du thèorème......Page 117
References......Page 121
1. Introduction......Page 124
2. Entropy of continuous and holomorphic maps......Page 126
3. Definitions of entropy......Page 128
4. Rational maps......Page 130
5. Entropy of rational maps......Page 133
6. Currents......Page 136
References......Page 137
1. Introduction......Page 140
2. Flows without fixed points......Page 141
3. Singular fixed points......Page 144
4. Functions with logarithmic singularities......Page 145
5. Some problems......Page 153
References......Page 154
1. Introduction......Page 156
2. Hyperbolic diffeomorphisms......Page 163
3. Flexibility......Page 169
4. Rigidity......Page 175
5. Hausdorff measures......Page 178
Acknowledgments......Page 185
References......Page 186
1. Introduction......Page 190
2. Main definitions......Page 194
3. Examples......Page 203
4. Reversibility and self-adjointness......Page 208
5. Ergodicity......Page 210
6. Spectrum......Page 213
7. Diffusion limit......Page 223
8. Remark about residence time in cells......Page 229
References......Page 231
1. Introduction......Page 234
2. Wang tiles: definitions and history......Page 236
3. Aperiodicity......Page 238
4. Existence of a valid tiling......Page 242
5. Tweaking the colors......Page 246
6. Generalization......Page 248
7. Mealy machine representation......Page 250
References......Page 251
1. Introduction......Page 254
2. Cut and project schemes......Page 257
The spectrum of a cut and project schemes......Page 259
The discrete case......Page 261
Realization in a large class of groups......Page 262
Properties of model sets......Page 266
5. Model set dynamical systems......Page 269
6. Realization for model sets......Page 276
7. The diffraction spectrum......Page 280
References......Page 281
CONTENTS......Page 284
2. Smooth realization of measure-preserving maps (Anatole Katok)......Page 285
3. Coexistence of KAM circles and positive entropy in area-preserving twist maps (presented by Anatole Katok)......Page 286
4. Orbit growth in polygonal billiards (Anatole Katok)......Page 288
6. Symbolic extensions (Michael Boyle and Sheldon Newhouse)......Page 289
7. Measures of maximal entropy (presented by Sheldon Newhouse)......Page 294
9. Sinai–Ruelle–Bowen measures and natural measures (presented by Michał Misiurewicz)......Page 295
10. Billiards (Domokos Szasz)......Page 296
11. Stable ergodicity (with Keith Burns)......Page 297
12. Mixing in Anosov flows (Michael Field)......Page 299
13. The structure of hyperbolic sets (contributed by Todd Fisher)......Page 300
14. The dynamics of geodesic flows (presented by Gerhard Knieper)......Page 302
15. Averaging (Yuri Kifer)......Page 303
16. Classifying Anosov diffeomorphisms and actions (presented by Anatole Katok and Ralf Spatzier)......Page 305
17. Invariant measures for hyperbolic actions of higher-rank abelian groups (Anatole Katok)......Page 308
18. Rigidity of higher-rank abelian actions (presented by Danijela Damjanovic)......Page 309
19. Local rigidity of actions (presented by David Fisher)......Page 310
20. Smooth and geometric rigidity......Page 311
21. Quantitative symplectic geometry (Helmut Hofer)......Page 323
23. Foliations (presented by Steven Hurder)......Page 325
24. “Fat” self-similar sets (Mark Pollicott)......Page 326
References......Page 327