ISBN: 3-540-66316-9
TITLE: Dynamical Systems, Ergodic Theory and Applications
AUTHOR: Bunimovich, L.A.; Dani, S.G.; Dobrushin, R.L.; Jakobson, M.V.; Kornfeld, I.P.; Maslova, N.B.; Pesin, Ya.B.; Sinai, Ya.G.; Smillie, J.; Sukhov, Yu.M.; Vershik, A.M.
TOC:

I. General Ergodic Theory of Groups of Measure Preserving Transformations Contents
Chapter 1. Basic Notions of Ergodic Theory and Examples of Dynamical Systems (I.P. Kornfeld, Ya.G. Sinai) 2
 1. Dynamical Systems with Invariant Measures 2
 2. First Corollaries of the Existence of Invariant Measures. Ergodic Theorems 11
 3. Ergodicity. Decomposition into Ergodic Components. Various Mixing Conditions 18
 4. General Constructions 23
4.1. Direct Products of Dynamical Systems 23
4.2. Skew Products of Dynamical Systems 24
4.3. Factor-Systems 25
4.4. Integral and Induced Automorphisms 25
4.5. Special Flows and Special Representations of Flows 26
4.6. Natural Extensions of Endomorphisms 28
Chapter 2. Spectral Theory of Dynamical Systems (I.P. Kornfeld, Ya.G. Sinai) 30
 1. Groups of Unitary Operators and Semigroups of Isometric Operators Adjoint to Dynamical Systems 30
 2. The Structure of the Dynamical Systems with Pure Point and Quasidiscrete Spectra 33
 3. Examples of Spectral Analysis of Dynamical Systems 35
 4. Spectral Analysis of Gauss Dynamical Systems 36
Chapter 3. Entropy Theory of Dynamical Systems (I.P. Kornfeld, Ya.G. Sinai) 38
 1. Entropy and Conditional Entropy of a Partition 39
 2. Entropy of a Dynamical System 40
 3. The Structure of Dynamical Systems of Positive Entropy 43
 4. The Isomorphy Problem for Bernoulli Automorphisms and K-Systems 45
 5. Equivalence of Dynamical Systems in the Sense of Kakutani 53
 6. Shifts in the Spaces of Sequences and Gibbs Measures 57
Chapter 4. Periodic Approximations and Their Applications. Ergodic Theorems, Spectral and Entropy Theory for the General Group Actions (I.P. Kornfeld, A.M. Vershik) 61
 1. Approximation Theory of Dynamical Systems by Periodic Ones. Flows on the Two-Dimensional Torus 61
 2. Flows on the Surfaces of Genus p 1 and Interval Exchange Transformations 66
 3. General Group Actions 69
3.1. Introduction 69
3.2. General Definition of the Actions of Locally Compact Groups on Lebesgue Spaces 70
3.3. Ergodic Theorems 71
3.4. Spectral Theory 74
 4. Entropy Theory for the Actions of General Groups 76
Chapter 5. Trajectory Theory (A.M. Vershik) 80
 1. Statements of Main Results 80
 2. Sketch of the Proof. Tame Partitions 84
 3. Trajectory Theory for Amenable Groups 89
 4. Trajectory Theory for Non-Amenable Groups. Rigidity 91
 5. Concluding Remarks. Relationship Between Trajectory Theory and Operator Algebras 94
Bibliography 95
Additional Bibliography 101
II. Ergodic Theory of Smooth Dynamical Systems
Chapter 6. Stochasticity of Smooth Dynamical Systems. The Elements of KAM-Theory (Ya.G. Sinai) 106
 1. Integrable and Nonintegrable Smooth Dynamical Systems. The Hierarchy of Stochastic Properties of Deterministic Dynamics 106
 2. The Kolmogorov-Arnold-Moser Theory (KAM-Theory) 109
Chapter 7. General Theory of Smooth Hyperbolic Dynamical Systems (Ya.B. Pesin) 113
 1. Hyperbolicity of Individual Trajectories 113
1.1. Introductory Remarks 113
1.2. Uniform Hyperbolicity 114
1.3. Nonuniform Hyperbolicity 115
1.4. Local Manifolds 116
1.5. Global Manifolds 118
 2. Basic Classes of Smooth Hyperbolic Dynamical Systems. Definitions and Examples 118
2.1. Anosov Systems 118
2.2. Hyperbolic Sets 121
2.3. Locally Maximal Hyperbolic Sets 124
2.4. Axiom A-Diffeomorphisms 125
2.5. Hyperbolic Attractors. Repellers 126
2.6. Partially Hyperbolic Dynamical Systems 128
2.7. Mather Theory 129
2.8. Nonuniformely Hyperbolic Dynamical Systems. Lyapunov Exponents 131
 3. Ergodic Properties of Smooth Hyperbolic Dynamical Systems 133
3.1. u-Gibbs Measures 133
3.2. Symbolic Dynamics 135
3.3. Measures of Maximal Entropy 137
3.4. Construction of u-Gibbs Measures 137
3.5. Topological Pressure and Topological Entropy 138
3.6. Properties of u-Gibbs Measures 141
3.7. Small Stochastic Perturbations 142
3.8. Equilibrium States and Their Ergodic Properties 143
3.9. Ergodic Properties of Dynamical Systems with Nonzero Lyapunov Exponents 144
3.10.Ergodic Properties of Anosov Systems and of UPH-Systems 146
3.11.Continuous Time Dynamical Systems 149
 4. Hyperbolic Geodesic Flows 149
4.1. Manifolds with Negative Curvature 149
4.2. Riemannian Metrics Without Conjugate (or Focal) Points 153
4.3. Entropy of Geodesic Flows 156
4.4. Riemannian Metrics of Nonpositive Curvature 157
 5. Geodesic Flows on Manifolds with Constant Negative Curvature 158
 6. Dimension-like Characteristics of Invariant Sets for Dynamical Systems 161
6.1. Introductory Remarks 161
6.2. Hausdorff Dimension 161
6.3. Other Dimension Characteristics 164
6.4. Carathodory Dimension Structure. Carathodory Dimension Characteristics 167
6.5. Examples of C-structures and Carathodory Dimension Characteristics 169
6.6. Multifractal Formalism 176
 7. Coupled Map Lattices 182
Additional References 190
Chapter 8. Billiards and Other Hyperbolic Systems (L.A. Bunimovich) 192
 1. Billiards 192
1.1. The General Definition of a Billiard 192
1.2. Billiards in Polygons and Polyhedrons 194
1.3. Billiards in Domains with Smooth Convex Boundary 196
1.4. Dispersing or Sinai Billiards 198
1.5. The Lorentz Gas and Hard Spheres Gas 206
1.6. Semi-dispersing Billiards and Boltzmann Hypotheses 206
1.7. Billiards in Domains with Boundary Possessing Focusing Components 209
1.8. Hyperbolic Dynamical Systems with Singularities (a General Approach) 215
1.9. Markov Approximations and Symbolic Dynamics for Hyperbolic Billiards 217
1.10.Statistical Properties of Dispersing Billiards and of the Lorentz Gas 219
1.11.Transport Coefficients for the Simplest Mechanical Models 222
 2. Strange Attractors 224
2.1. Definition of a Strange Attractor 224
2.2. The Lorenz Attractor 225
2.3. Some Other Examples of Hyperbolic Strange Attractors 230
Additional References 231
Chapter 9. Ergodic Theory of One-Dimensional Mappings (M.V. Jakobson) 234
 1. Expanding Maps 234
1.1. Definitions, Examples, the Entropy Formula 234
1.2. Walters Theorem 237
 2. Absolutely Continuous Invariant Measures for Nonexpanding Maps 239
2.1. Some Examples 239
2.2. Intermittency of Stochastic and Stable Systems 241
2.3. Ergodic Properties of Absolutely Continuous Invariant Measures 243
 3. Feigenbaum Universality Law 245
3.1. The Phenomenon of Universality 245
3.2. Doubling Transformation 247
3.3. Neighborhood of the Fixed Point 249
3.4. Properties of Maps Belonging to the Stable Manifold of 251
 4. Rational Endomorphisms of the Riemann Sphere 252
4.1. The Julia Set and Its Complement 252
4.2. The Stability Properties of Rational Endomorphisms 254
4.3. Ergodic and Dimensional Properties of Julia Sets 255
Bibliography 256
III. Dynamical Systems on Homogeneous Spaces
Chapter 10. Dynamical Systems on Homogeneous Spaces (S.G. Dani) 266
 1. Introduction 266
1.1. Measures on homogeneous spaces 266
1.2. Examples of lattices 268
1.3. Ergodicity and its consequences 271
1.4. Isomorphisms and factors of affine automorphisms 272
 2. Affine automorphisms of tori and nilmanifolds 273
2.1. Ergodic properties; the case of tori 273
2.2. Ergodic properties on nilmanifolds 275
2.3. Unipotent affine automorphisms 278
2.4. Quasi-unipotent affine automorphisms 280
2.5. Closed invariant sets of automorphisms 281
2.6. Dynamics of hyperbolic automorphisms 281
2.7. More on invariant sets of hyperbolic toral automorphisms 283
2.8. Distribution of orbits of hyperbolic automorphisms 285
2.9. Dynamics of ergodic toral automorphisms 286
2.10.Actions of groups of affine automorphisms 287
 3. Group-induced translation flows; special cases 289
3.1. Flows on solvmanifolds 289
3.2. Homogeneous spaces of semisimple groups 292
3.3. Flows on low-dimensional homogeneous spaces 295
 4. Ergodic properties of flows on general homogeneous spaces 297
4.1. Horospherical subgroups and Mautner phenomenon 298
4.2. Ergodicity of one-parameter flows 300
4.3. Invariant functions and ergodic decomposition 301
4.4. Actions of subgroups 303
4.5. Duality 304
4.6. Spectrum and mixing of group-induced flows 305
Contents 265
4.7. Mixing of higher orders 306
4.8. Entropy 307
4.9. K-mixing, Bernoullicity 308
 5. Group-induced flows with hyperbolic structure 309
5.1. Anosov automorphisms 309
5.2. Affine automorphisms with a hyperbolic fixed point 311
5.3. Anosov flows 312
 6. Invariant measures of group-induced flows 313
6.1. Invariant measures of Ad-unipotent flows 313
6.2. Invariant measures and epimorphic subgroups 316
6.3. Invariant measures of actions of diagonalisable groups 318
6.4. A weak recurrence property and infinite invariant measures 318
6.5. Distribution of orbits and polynomial trajectories 320
6.6. A uniform version of uniform distribution 321
6.7. Distribution of translates of closed orbits 323
 7. Orbit closures of group-induced flows 323
7.1. Homogeneity of orbit closures 323
7.2. Orbit closures of horospherical subgroups 325
7.3. Orbits of reductive subgroups 327
7.4. Orbit closures of one-parameter flows 328
7.5. Dense orbits and minimal sets of flows 330
7.6. Divergent trajectories of flows 332
7.7. Bounded orbits and escapable sets 333
 8. Duality and lattice-actions on vector spaces 335
8.1. Duality between orbits 335
8.2. Duality of invariant measures 336
 9. Applications to Diophantine approximation 338
9.1. Polynomials in one variable 338
9.2. Values of linear forms 338
9.3. Diophantine approximation with dependent quantities 339
9.4. Values of quadratic forms 340
9.5. Forms of higher degree 343
9.6. Integral points on algebraic varieties 343
 10. Classification and related questions 344
10.1.Metric isomorphisms and factors 345
10.2.Metric rigidity 346
10.3.Topological conjugacy 347
10.4.Topological equivalence 349
Bibliography 350
IV. The Dynamics of Billiard Flows in Rational Polygons
Chapter 11. The Dynamics of Billiard Flows in Rational Polygons of Dynamical Systems (J. Smillie) 360
 1. Two Examples 362
 2. Formal Properties of the Billiard Flow 364
 3. The Flow in a Fixed Direction 367
 4. Billiard Techniques: Minimality and Closed Orbits 369
 5. Billiard Techniques: Unique Ergodicity 372
 6. Dynamics on Moduli Spaces 374
 7. The Lattice Examples of Veech 377
Bibliography 380
V. Dynamical Systems of Statistical Mechanics and Kinetic Equations
Chapter 12. Dynamical Systems of Statistical Mechanics (R.L. Dobrushin, Ya.G. Sinai, Yu.M. Sukhov) 384
 1. Introduction 384
 2. Phase Space of Systems of Statistical Mechanics and Gibbs Measures 386
2.1. The Configuration Space 386
2.2. Poisson Measures 388
2.3. The Gibbs Configuration Probability Distribution 388
2.4. Potential of the Pair Interaction. Existence and Uniqueness of a Gibbs Configuration Probability Distribution 390
2.5. The Phase Space. The Gibbs Probability Distribution 393
2.6. Gibbs Measures with a General Potential 395
2.7. The Moment Measure and Moment Function 396
 3. Dynamics of a System of Interacting Particles 398
3.1. Statement of the Problem 398
3.2. Construction of the Dynamics and Time Evolution 400
3.3. Hierarchy of the Bogolyubov Equations 402
 4. Equilibrium Dynamics 403
4.1. Definition and Construction of Equilibrium Dynamics 403
4.2. The Gibbs Postulate 405
4.3. Degenerate Models 407
4.4. Asymptotic Properties of the Measures Pt 408
 5. Ideal Gas and Related Systems 408
5.1. The Poisson Superstructure 408
5.2. Asymptotic Behaviour of the Probability Distribution Pt as t 410
5.3. The Dynamical System of One-Dimensional Hard Rods 411
 6. Kinetic Equations 412
6.1. Statement of the Problem 412
6.2. The Boltzmann Equation 415
6.3. The Vlasov Equation 419
6.4. The Landau Equation 420
6.5. Hydrodynamic Equations 421
Bibliography 423
Chapter 13. Existence and Uniqueness Theorems for the Boltzmann Equation (N.B. Maslova) 430
 1. Formulation of Boundary Problems. Properties of Integral Operators 430
1.1. The Boltzmann Equation 430
1.2. Formulation of Boundary Problems 434
1.3. Properties of the Collision Integral 435
 2. Linear Stationary Problems 437
2.1. Asymptotics 437
2.2. Internal Problems 438
2.3. External Problems 439
2.4. Kramers' Problem 441
 3. Nonlinear Stationary Problems 441
 4. Non-Stationary Problems 443
4.1. Relaxation in a Homogeneous Gas 443
4.2. The Cauchy Problem 444
4.3. Boundary Problems 445
 5. On a Connection of the Boltzmann Equation with Hydrodynamic Equations 446
5.1. Statement of the Problem 446
5.2. Local Solutions. Reduction to Euler Equations 448
5.3. A Global Theorem. Reduction to Navier-Stokes Equations 450
Bibliography 452
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