Dynamical Systems II: Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics

✍ Scribed by Ya. G. Sinai

• Publisher: Springer-Verlag
• Year: 1989
• Tongue: English
• Leaves: 286
• Edition: 1
• Category: Library

✦ Synopsis

Following the concept of the EMS series this volume sets out to familiarize the reader to the fundamental ideas and results of modern ergodic theory and to its applications to dynamical systems and statistical mechanics. The exposition starts from the basic of the subject, introducing ergodicity, mixing and entropy. Then the ergodic theory of smooth dynamical systems is presented - hyperbolic theory, billiards, one-dimensional systems and the elements of KAM theory. Numerous examples are presented carefully along with the ideas underlying the most important results. The last part of the book deals with the dynamical systems of statistical mechanics, and in particular with various kinetic equations. This book is compulsory reading for all mathematicians working in this field, or wanting to learn about it.

✦ Table of Contents

Cover
Title: Dynamical System II: Ergodic Theory with Applications to dynamical Systems and Statistical Mechanics
Encyclopaedia of Mathematical Sciences, Volume 2
Preface
Contents
I. General Ergodic Theory of Groups ofMeasure Preserving Transformations
Contents
Chapter 1: Basic Notions of Ergodic Theory and Examples of Dynamical Systems
§ 1. Dynamical Systems with Invariant Measures
§2. First Corollaries of the Existence of Invariant Measures. Ergodic Theorems
§3. Ergodicity. Decomposition into Ergodic Components. Various Mixing Conditions
§4. General Constructions
4.1. Direct Products of Dynamical Systems
4.2. Skew Products of Dynamical Systems
4.3. Factor-Systems
4.4. Integral and Induced Automorphisms
4.5. Special Flows and Special Representations of Flows
4.6. Natural Extensions of Endomo
Chapter 2: Spectral Theory of Dynamical Systems
§ 1. Groups of Unitary Operators and Semigroups of Isometric Operators Adjoint to Dynamical Systems
§2. The Structure of the Dynamical Systems with Pure Point and Quasidiscrete Spectra
§3. Examples of Spectral Analysis of Dynamical Systems
§4. Spectral Analysis of Gauss Dynamical Systems
Chapter 3: Entropy Theory of Dynamical Systems
§ 1. Entropy and Conditional Entropy of a Partition
§2. Entropy of a Dynamical System
§3. The Structure of Dynamical Systems of Positive Entropy
§4. The Isomorphy Problem for Bernoulli Automorphisms and K-Systems
§5. Equivalence of Dynamical Systems in the Sense of Kakutani
§6. Shifts in the Spaces of Sequences and Gibbs Measures
Chapter 4: Periodic Approximations and Their Applications. Ergodic Theorems, Spectral and Entropy Theory for the General Group Actions!
§ 1. Approximation Theory of Dynamical Systems by Periodic Ones. Flows on the Two-Dimensional Torus
§2. Flows on the Surfaces of Genus p>1 and Interval Exchange Transformations
§3. General Group Actions
3.1. Introduction
3.2. General Definition of the Actions of Locally Compact Groups on Lebesgue Spaces.
3.3. Ergodic Theorems
§4. Entropy Theory for the Actions of General Groups
Chapter 5: Trajectory Theory
§1. Statements of Main Results
§2. Sketch of the Proof. Tame Partitions
§3.Trajectory Theory for Amenable Groups
§4. Trajectory Theory for Non-Amenable Groups. Rigidity
§ 5. Concluding Remarks. Relationship Between Trajectory Theory and Operator Algebras
Bibliography
II. Ergodic Theory of Smooth Dynamical Systems
Contents
Chapter 6: Stochasticity of Smooth Dynamical Systems. The Elements of KAM-Theory
§ 1. Integrable and Nonintegrable Smooth Dynamical Systems. The Hierarchy of Stochastic Properties of Deterministic Dynamics
§2. The Kolmogorov-Arnold-Moser Theory (KAM-Theory)
Chapter 7: General Theory of Smooth Hyperbolic Dynamical Systems
§ 1. Hyperbolicity of Individual Trajectories
1.1. Introductory Remarks
1.2. Uniform Hyperbolicity
1.3. Nonuniform Hyperbolicity
1.4. Local Manifolds
1.5. Global Manifolds
§2. Basic Classes of Smooth Hyperbolic Dynamical Systems. Definitions and Examples
2.1. Anosov Systems
2.2. Hyperbolic Sets
2.3. Locally Maximal Hyperbolic Sets
2.4. A-Diffeomorphisms
2.5. Hyperbolic Attractors
2.6. Partially Hyperbolic Dynamical Systems
2.7. Mather Theory
2.8. Nonuniformely Hyperbolic Dynamical Systems. Lyapunov Exponents
§3. Ergodic Properties of Smooth Hyperbolic Dynamical Systems
3.1. u-Gibbs Measures
3.2. Symbolic Dynamics
3.3. Measures of Maximal Entropy
3.4. Construction of u-Gibbs Measures
3.5. Topological Pressure and Topological Entropy
3.6. Properties of u-Gibbs Measures
3.7. Small Stochastic Perturbations
3.8. Equilibrium States and Their Ergodic Properties
3.9. Ergodic Properties of Dynamical Systems with Nonzero Lyapunov Exponents.
3.10. Ergodic Properties of Anosov Systems and ofUPH-Systems
3.11. Continuous Time Dynamical Systems
§4. Hyperbolic Geodesic Flows
4.1. Manifolds with Negative Curvature
4.2. Riemannian Metrics Without Conjugate Points
4.3. Entropy of Geodesic Flow
§ 5. Geodesic Flows on Manifolds with Constant Negative Curvature
§6. Dimension-like Characteristics of Invariant Sets for Dynamical Systems
6.1. Introductory Remarks
6.2. Hausdorff Dimension
6.3. Dimension with Respect to a Dynamical System
6.4. Capacity and Other Characteristics
Chapter 8: Dynamical Systems of Hyperbolic Type with Singularities
§ 1. Billiards
1.1. The General Definition of a Billiard
1.2. Billiards in Polygons and Polyhedrons
1.3. Billiards in Domains with Smooth Convex Boundary
1.4. Dispersing (Sinai) Billiards
1.5. The Lorentz Gas and the Hard Spheres Gas
1.6. Semidispersing Billiards
1.7. Billiards in Domains with Boundary Possessing Focusing Components
1.8. Hyperbolic Dynamical Systems with Singularities (a General Approach).
1.9. The Markov Partition and Symbolic Dynamics for Dispersing Billiards
1.10. Statistical Properties of Dispersing Billiards and of the Lorentz Gas
§2. Strange Attractors
2.1. Definition of a Strange Attractor
2.2. The Lorenz Attractor
2.3. Some Other Examples of Hyperbolic Strange Attractors
Chapter 9: Ergodic Theory of One-Dimensional Mappings
§ 1. Expanding Maps
1.1. Definitions, Examples, the Entropy Formula
1.2. Walters Theorem
§2. Absolutely Continuous Invariant Measures for Nonexpanding Maps
2.1. Some Examples
2.2. Intermittency of Stochastic and Stable Systems
2.3. Ergodic Properties of Absolutely Continuous Invariant Measures
§ 3. Feigenbaum Universality Law
3.1. The Phenomenon of Universality
3.2. Doubling Transformation
3.3. Neighborhood of the Fixed Point
3.4. Properties of Maps Belonging to the Stable Manifold of tP
§4. Rational Endomorphisms of the Riemann Sphere
4.1. The Julia Set and its Complement
4.2. The Stability Properties of Rational Endomorphisms
4.3. Ergodic and Dimensional Properties of Julia Sets.
Bibliography
III. Dynamical Systems of Statistical Mechanics and Kinetic Equations
Contents
Chapter 10: Dynamical Systems of Statistical Mechanics
§ 1. Introduction
§2. Phase Space of Systems of Statistical Mechanics and Gibbs Measures
2.1. The Configuration Space
2.2. Poisson Measures
2.3. The Gibbs Configuration Probability Distribution
2.4. Potential of the Pair Interaction. Existence and Uniqueness of a Gibbs Configuration Probability Distribution
2.5. The Phase Space. The Gibbs Probability Distribution
2.6. Gibbs Measures with a General Potential
2.7. The Moment Measure and Moment Function
§3. Dynamics of a System of Interacting Particles
3.1. Statement of the Problem
3.2. Construction of the Dynamics and Time Evolution.
3.3. Hierarchy of the Bogolyubov Equations
§4. Equilibrium Dynamics
4.1. Definition and Construction of Equilibrium Dynamics
4.2. The Gibbs Postulate
4.3. Degenerate Models
4.4. Asymptotic Properties 6f the Measures Pt.
§5. Ideal Gas and Related Systems
5.1. The Poisson Superstructure
5.2. Asymptotic Behaviour of the Probability Distribution Pt as t--> inf
5.3. The Dynamical System of One-Dimensional Hard Rods
§6. Kinetic Equations
6.1. Statement of the Problem
6.2. The Boltzmann Equation
6.3. The Vlasov Equation
6.4. The Landau Equation
6.5. Hydrodynamic Equations
Bibliography
Chapter 11: Existence and Uniqueness Theorems for the Boltzmann Equation
§ 1. Formulation of Boundary Problems. Properties of Integral Operators
1.1. The Boltzmann Equation.
1.2. Formulation of Boundary Problems
1.3. Properties of the Collision Integral
§2. Linear Stationary Problems
2.1. Asymptotics
2.2. Internal Problems
2.3. External Problems
2.4. Kramers' Problem
§3. Nonlinear Stationary Problems
§4. Non-Stationary Problems
4.1. Relaxation in a Homogeneous Gas
4.2. The Cauchy Problem
4.3. Boundary Problems
§ 5. On a Connection of the Boltzmann Equation with Hydrodynamic Equations
5.1. Statement of the Problem
5.2. Local Solutions. Reduction to Euler Equations.
5.3. A Global Theorem. Reduction to Navier-Stokes Equations
Bibliography
Subject Index