By: L.A. Bunimovich, Yu.M. Sukhov, Anatoly M. Vershik, S.G. Dani, R.L. Dobrushin

Hardcover

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This EMS volume, the first edition of which was published as Dynamical Systems II, EMS 2, sets out to familiarize the reader to the fundamental ideas and results of modern ergodic theory and its applications to dynamical systems and statistical mechanics. The exposition starts from the basic of the subject, introducing ergodicity, mixing and entropy. The ergodic theory of smooth dynamical systems is treated. Numerous examples are presented carefully along with the ideas underlying the most important results. Moreover, the book deals with the dynamical systems of statistical mechanics, and with various kinetic equations. For this second enlarged and revised edition, published as Mathematical Physics I, EMS 100, two new contributions on ergodic theory of flows on homogeneous manifolds and on methods of algebraic geometry in the theory of interval exchange transformations were added. This book is compulsory reading for all mathematicians working in this field, or wanting to learn about it.

"... The list of topics gives some idea of the impressive scope of this volume, and the comprehensive bibliographies represent the state of knowledge right up to the late 1990's. ... This survey is much more than a place where one can quickly look up the current state of knowledge on a particular topic or get an idea about the scope of a branch of the theroy. It is also highly recommended and fascinating reading for experts in the subject: the individual chapters are written by top experts inthe field, whose insights and illuminating remarks will reward readers at any level of expertise. The coverage of both 'mathematical' and 'physical' aspects of the theory in a single volume and from a reasonably unified point of view is another very attractive and quite unique feature of this volume." K.Schmidt, Wien, Monathshefte fur Mathematik, Vol. 139, Issue 4, p.351, 2003 "... This Encyclopaedia volume is indeed (as the publisher writes on the back-cover of the book) compulsory reading for all mathematicians working in this field, or wanting to learn about it." W.T.van Horssen, ZAMM 82 (2002), Issue 8

General Ergodic Theory of Groupsof Measure Preserving Transformations	p. 1
Ergodic Theory of Smooth Dynamical Systems	p. 103
Dynamical Systems on Homogeneous Spaces	p. 264
The Dynamics of Billiard Flowsin Rational Polygons	p. 360
Dynamical Systemsof Statistical Mechanics and Kinetic Equations	p. 383
Subject Index	p. 455
General Ergodic Theory of Groupsof Measure Preserving Transformations
Basic Notions of Ergodic Theory and Examples of Dynamical Systems	p. 2
Dynamical Systems with Invariant Measures	p. 2
First Corollaries of the Existence of Invariant Measures. Ergodic Theorems	p. 11
Ergodicity. Decomposition into Ergodic Components. Various Mixing Conditions	p. 18
General Constructions	p. 23
Direct Products of Dynamical Systems	p. 23
Skew Products of Dynamical Systems	p. 24
Factor-Systems	p. 25
Integral and Induced Automorphisms	p. 25
Special Flows and Special Representations of Flows	p. 26
Natural Extensions of Endomorphisms	p. 28
Spectral Theory of Dynamical Systems	p. 30
Groups of Unitary Operators and Semigroups of Isometric Operators Adjoint to Dynamical Systems	p. 30
The Structure of the Dynamical Systems with Pure Point and Quasidiscrete Spectra	p. 33
Examples of Spectral Analysis of Dynamical Systems	p. 35
Spectral Analysis of Gauss Dynamical Systems	p. 36
Entropy Theory of Dynamical Systems	p. 38
Entropy and Conditional Entropy of a Partition	p. 39
Entropy of a Dynamical System	p. 40
The Structure of Dynamical Systems of Positive Entropy	p. 43
The Isomorphy Problem for Bernoulli Automorphisms and K-Systems	p. 45
Equivalence of Dynamical Systems in the Sense of Kakutani	p. 53
Shifts in the Spaces of Sequences and Gibbs Measures	p. 57
Periodic Approximations and Their Applications. Ergodic Theorems, Spectral and Entropy Theory for the General Group Actions	p. 61
Approximation Theory of Dynamical Systems by Periodic Ones. Flows on the Two-Dimensional Torus	p. 61
Flows on the Surfaces of Genus p ≥ 1 and Interval Exchange Transformations	p. 66
General Group Actions	p. 69
Introduction	p. 69
General Definition of the Actions of Locally Compact Groups on Lebesgue Spaces	p. 70
Ergodic Theorems	p. 71
Spectral Theory	p. 74
Entropy Theory for the Actions of General Groups	p. 76
Trajectory Theory	p. 80
Statements of Main Results	p. 80
Sketch of the Proof. Tame Partitions	p. 84
Trajectory Theory for Amenable Groups	p. 89
Trajectory Theory for Non-Amenable Groups. Rigidity	p. 91
Concluding Remarks. Relationship Between Trajectory Theory and Operator Algebras	p. 94
Bibliography	p. 95
Additional Bibliography	p. 101
Ergodic Theory of SmoothDynamical Systems
Stochasticity of Smooth Dynamical Systems. The Elements of KAM-Theory	p. 106
Integrable and Nonintegrable Smooth Dynamical Systems. The Hierarchy of Stochastic Properties of Deterministic Dynamics	p. 106
The Kolmogorov-Arnold-Moser Theory (KAM-Theory)	p. 109
General Theory of Smooth Hyperbolic Dynamical Systems	p. 113
Hyperbolicity of Individual Trajectories	p. 113
Introductory Remarks	p. 113
Uniform Hyperbolicity	p. 114
Nonuniform Hyperbolicity	p. 115
Local Manifolds	p. 116
Global Manifolds	p. 118
Basic Classes of Smooth Hyperbolic Dynamical Systems. Definitions and Examples	p. 118
Anosov Systems	p. 118
Hyperbolic Sets	p. 121
Locally Maximal Hyperbolic Sets	p. 124
Axiom A-Diffeomorphisms	p. 125
Hyperbolic Attractors. Repellers	p. 126
Partially Hyperbolic Dynamical Systems	p. 128
Mather Theory	p. 129
Nonuniformely Hyperbolic Dynamical Systems. Lyapunov Exponents	p. 131
Ergodic Properties of Smooth Hyperbolic Dynamical Systems	p. 133
u-Gibbs Measures	p. 133
Symbolic Dynamics	p. 135
Measures of Maximal Entropy	p. 137
Construction of u-Gibbs Measures	p. 137
Topological Pressure and Topological Entropy	p. 138
Properties of u-Gibbs Measures	p. 141
Small Stochastic Perturbations	p. 142
Equilibrium States and Their Ergodic Properties	p. 143
Ergodic Properties of Dynamical Systems with Nonzero Lyapunov Exponents	p. 144
Ergodic Properties of Anosov Systems and of UPH-Systems	p. 146
Continuous Time Dynamical Systems	p. 149
Hyperbolic Geodesic Flows	p. 149
Manifolds with Negative Curvature	p. 149
Riemannian Metrics Without Conjugate (or Focal) Points	p. 153
Entropy of Geodesic Flows	p. 156
Riemannian Metrics of Nonpositive Curvature	p. 157
Geodesic Flows on Manifolds with Constant Negative Curvature	p. 158
Dimension-like Characteristics of Invariant Sets for Dynamical Systems	p. 161
Introductory Remarks	p. 161
Hausdorff Dimension	p. 161
Other Dimension Characteristics	p. 164
Carathéodory Dimension Structure. Carathéodory Dimension Characteristics	p. 167
Examples of C-structures and Carathéodory Dimension Characteristics	p. 169
Multifractal Formalism	p. 176
Coupled Map Lattices	p. 182
Additional References	p. 190
Billiards and Other Hyperbolic Systems	p. 192
Billiards	p. 192
The General Definition of a Billiard	p. 192
Billiards in Polygons and Polyhedrons	p. 194
Billiards in Domains with Smooth Convex Boundary	p. 196
Dispersing or Sinai Billiards	p. 198
The Lorentz Gas and Hard Spheres Gas	p. 206
Semi-dispersing Billiards and Boltzmann Hypotheses	p. 206
Billiards in Domains with Boundary Possessing Focusing Components	p. 209
Hyperbolic Dynamical Systems with Singularities (a General Approach)	p. 215
Markov Approximations and Symbolic Dynamics for Hyperbolic Billiards	p. 217
Statistical Properties of Dispersing Billiards and of the Lorentz Gas	p. 219
Transport Coefficients for the Simplest Mechanical Models	p. 222
Strange Attractors	p. 224
Definition of a Strange Attractor	p. 224
The Lorenz Attractor	p. 225
Some Other Examples of Hyperbolic Strange Attractors	p. 230
Additional References	p. 231
Ergodic Theory of One-Dimensional Mappings	p. 234
Expanding Maps	p. 234
Definitions, Examples, the Entropy Formula	p. 234
Walters Theorem	p. 237
Absolutely Continuous Invariant Measures for Nonexpanding Maps	p. 239
Some Examples	p. 239
Intermittency of Stochastic and Stable Systems	p. 241
Ergodic Properties of Absolutely Continuous Invariant Measures	p. 243
Feigenbaum Universality Law	p. 245
The Phenomenon of Universality	p. 245
Doubling Transformation	p. 247
Neighborhood of the Fixed Point	p. 249
Properties of Maps Belonging to the Stable Manifold of ¿	p. 251
Rational Endomorphisms of the Riemann Sphere	p. 252
The Julia Set and Its Complement	p. 252
The Stability Properties of Rational Endomorphisms	p. 254
Ergodic and Dimensional Properties of Julia Sets	p. 255
Bibliography	p. 256
Dynamical Systemson Homogeneous Spaces
Dynamical Systems on Homogeneous Spaces	p. 266
Introduction	p. 266
Measures on homogeneous spaces	p. 266
Examples of lattices	p. 268
Ergodicity and its consequences	p. 271
Isomorphisms and factors of affine automorphisms	p. 272
Affine automorphisms of tori and nilmanifolds	p. 273
Ergodic properties; the case of tori	p. 273
Ergodic properties on nilmanifolds	p. 275
Unipotent affine automorphisms	p. 278
Quasi-unipotent affine automorphisms	p. 280
Closed invariant sets of automorphisms	p. 281
Dynamics of hyperbolic automorphisms	p. 281
More on invariant sets of hyperbolic toral automorphisms	p. 283
Distribution of orbits of hyperbolic automorphisms	p. 285
Dynamics of ergodic toral automorphisms	p. 286
Actions of groups of affine automorphisms	p. 287
Group-induced translation flows; special cases	p. 289
Flows on solvmanifolds	p. 289
Homogeneous spaces of semisimple groups	p. 292
Flows on low-dimensional homogeneous spaces	p. 295
Ergodic properties of flows on general homogeneous spaces	p. 297
Horospherical subgroups and Mautner phenomenon	p. 298
Ergodicity of one-parameter flows	p. 300
Invariant functions and ergodic decomposition	p. 301
Actions of subgroups	p. 303
Duality	p. 304
Spectrum and mixing of group-induced flows	p. 305
Mixing of higher orders	p. 306
Entropy	p. 307
K-mixing, Bernoullicity	p. 308
Group-induced flows with hyperbolic structure	p. 309
Anosov automorphisms	p. 309
Affine automorphisms with a hyperbolic fixed point	p. 311
Anosov flows	p. 312
Invariant measures of group-induced flows	p. 313
Invariant measures of Ad-unipotent flows	p. 313
Invariant measures and epimorphic subgroups	p. 316
Invariant measures of actions of diagonalisable groups	p. 318
A weak recurrence property and infinite invariant measures	p. 318
Distribution of orbits and polynomial trajectories	p. 320
A uniform version of uniform distribution	p. 321
Distribution of translates of closed orbits	p. 323
Orbit closures of group-induced flows	p. 323
Homogeneity of orbit closures	p. 323
Orbit closures of horospherical subgroups	p. 325
Orbits of reductive subgroups	p. 327
Orbit closures of one-parameter flows	p. 328
Dense orbits and minimal sets of flows	p. 330
Divergent trajectories of flows	p. 332
Bounded orbits and escapable sets	p. 333
Duality and lattice-actions on vector spaces	p. 335
Duality between orbits	p. 335
Duality of invariant measures	p. 336
Applications to Diophantine approximation	p. 338
Polynomials in one variable	p. 338
Values of linear forms	p. 338
Diophantine approximation with dependent quantities	p. 339
Values of quadratic forms	p. 340
Forms of higher degree	p. 343
Integral points on algebraic varieties	p. 343
Classification and related questions	p. 344
Metric isomorphisms and factors	p. 345
Metric rigidity	p. 346
Topological conjugacy	p. 347
Topological equivalence	p. 349
Bibliography	p. 350
The Dynamics of Billiard Flowsin Rational Polygons
The Dynamics of Billiard Flows in Rational Polygons of Dynamical Systems	p. 360
Two Examples	p. 362
Formal Properties of the Billiard Flow	p. 364
The Flow in a Fixed Direction	p. 367
Billiard Techniques: Minimality and Closed Orbits	p. 369
Billiard Techniques: Unique Ergodicity	p. 372
Dynamics on Moduli Spaces	p. 374
The Lattice Examples of Veech	p. 377
Bibliography	p. 380
Dynamical Systems of Statistical Mechanicsand Kinetic Equations
Dynamical Systems of Statistical Mechanics	p. 384
Introduction	p. 384
Phase Space of Systems of Statistical Mechanics and Gibbs Measures	p. 386
The Configuration Space	p. 386
Poisson Measures	p. 388
The Gibbs Configuration Probability Distribution	p. 388
Potential of the Pair Interaction. Existence and Uniqueness of a Gibbs Configuration Probability Distribution	p. 390
The Phase Space. The Gibbs Probability Distribution	p. 393
Gibbs Measures with a General Potential	p. 395
The Moment Measure and Moment Function	p. 396
Dynamics of a System of Interacting Particles	p. 398
Statement of the Problem	p. 398
Construction of the Dynamics and Time Evolution	p. 400
Hierarchy of the Bogolyubov Equations	p. 402
Equilibrium Dynamics	p. 403
Definition and Construction of Equilibrium Dynamics	p. 403
The Gibbs Postulate	p. 405
Degenerate Models	p. 407
Asymptotic Properties of the Measures Pt	p. 408
Ideal Gas and Related Systems	p. 408
The Poisson Superstructure	p. 408
Asymptotic Behaviour of the Probability Distribution Pt as t → ∞	p. 410
The Dynamical System of One-Dimensional Hard Rods	p. 411
Kinetic Equations	p. 412
Statement of the Problem	p. 412
The Boltzmann Equation	p. 415
The Vlasov Equation	p. 419
The Landau Equation	p. 420
Hydrodynamic Equations	p. 421
Bibliography	p. 423
Existence and Uniqueness Theorems for the Boltzmann Equation	p. 430
Formulation of Boundary Problems. Properties of Integral Operators	p. 430
The Boltzmann Equation	p. 430
Formulation of Boundary Problems	p. 434
Properties of the Collision Integral	p. 435
Linear Stationary Problems	p. 437
Asymptotics	p. 437
Internal Problems	p. 438
External Problems	p. 439
Kramers' Problem	p. 441
Nonlinear Stationary Problems	p. 441
Non-Stationary Problems	p. 443
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540663164
ISBN-10: 3540663169
Series: Encyclopaedia of Mathematical Sciences
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 472
Published: July 2009
Dimensions (cm): 22.9 x 15.2  x 2.6
Weight (kg): 0.84