Ergodic Theory

I Ergodicity and Mixing. Examples of Dynamic Systems.- 1 Basic Definitions of Ergodic Theory.- x1. Definition of Dynamical Systems.- x2. The Birkhoff-Khinchin Ergodic Theorem. Ergodicity.- x3. Nonergodic Systems. Decomposition into Ergodic Components.- x4. Averaging in the Ergodic Case.- x5. Integral and Induced Automorphisms.- x6. Weak Mixing, Mixing, Multiple Mixing.- x7. Unitary and Isometric Operators Adjoint to Dynamical Systems.- x8. Dynamical Systems on Compact Metric Spaces.- 2 Smooth Dynamical Systems on Smooth Manifolds.- x1. Invariant Measures Compatible with Differentiability.- x2. Liouville's Theorem and the Dynamical Systems of Classical Mechanics.- x3. Integrable Dynamical Systems.- 3 Smooth Dynamical Systems on the Torus.- x1. Translations on the Torus.- x2. The Lagrange Problem.- x3. Homeomorphisms of the Circle.- x4. The Denjoy Theorem.- x5. Arnold's Example.- x6. The Ergodicity of Diffeomorphisms of the Circle with Respect to Lebesgue Measure.- 4 Dynamical Systems of Algebraic Origin.- x1. Translations on Compact Topological Groups.- x2. Skew Translations and Compound Skew Translations on Commutative Compact Groups.- x3. Endomorphisms and Automorphisms of Commutative Compact Groups.- x4. Dynamical Systems on Homogenous Spaces of the Group SL(2, ?).- 5 Interval Exchange Transformations.- x1. Definition of Interval Exchange Transformations.- x2. An Estimate of the Number of Invariant Measures.- x3. Absence of Mixing.- x4. An Example of a Minimal but not Uniquely Ergodic Interval Exchange Transformation.- 6 Billiards.- x1. The Construction of Dynamical Systems of the Billiards Type.- x2. Billiards in Polygons and Polyhedra.- x3. Billiards in Domains with Convex Boundary.- x4. Systems of One-dimensional Point-like Particles.- x5. Lorentz Gas and Systems of Hard Spheres.- 7 Dynamical Systems in Number Theory.- x1. Uniform Distribution.- x2. Uniform Distribution of Fractional Parts of Polynomials.- x3. Uniform Distribution of Fractional Parts of Exponential Functions.- x4. Ergodic Properties of Decompositions into Continuous Tractions and Piecewise-monotonic Maps.- 8 Dynamical Systems in Probability Theory.- x1. Stationary Random Processes and Dynamical Systems.- x2. Gauss Dynamical Systems.- 9 Examples of Infinite Dimensional Dynamical Systems.- x1. Ideal Gas.- x2. Dynamical Systems of Statistical Mechanics.- x3. Dynamical Systems and Partial Differential Equations.- II Basic Constructions of Ergodic Theory.- 10 Simplest General Constructions and Elements of Entropy Theory of Dynamical Systems.- x1. Direct and Skew Products of Dynamical Systems.- x2. Metric Isomorphism of Skew Products. Equivalence of Dynamical Systems in the Sense of Kakutani.- x3. Time Change in Flows.- x4. Endomorphisms and Their Natural Extensions.- x5. The Rohlin-Halmos Lemma.- x6. Entropy.- x7. Metric Isomorphism of Bernoulli Automorphisms.- x8. K-systems and Exact Endomorphisms.- 11 Special Representations of Flows.- x1. Definition of Special Flows.- x2. Statement of the Main Theorem on Special Representation of Flows and Examples of Special Flows.- x3. Proof of the Theorem on Special Representation.- x4. Rudolph's Theorem.- III Spectral Theory of Dynamical Systems.- 12 Dynamical Systems with Pure Point Spectrum.- x1. General Properties of Eigen-Values and Eigen-Functions of Dynamical Systems.- x2. Dynamical Systems with Pure Point Spectrum. The Case of Discrete Time.- x3. Dynamical Systems with Pure Point Spectrum. The Case of Continuous Time.- 13 Examples of Spectral Analysis of Dynamical Systems.- x1. Spectra of K-automorphisms.- x2. Spectra of Ergodic Automorphisms of Commutative Compact Groups.- x3. Spectra of Compound Skew Translations on the Torus and of Their Perturbations.- x4. Examples of the Spectral Analysis of Automorphisms with Singular Spectrum.- x5. Spectra of K-flows.- 14 Spectral Analysis of Gauss Dynamical Systems.- x1. The Decomposition of Hilbert Space L2(M, , u) into Hermite-Ito Polynomial Subspaces.- x2. Ergodicity and Mixing Criteria for Gauss Dynamical Systems.- x3. The Maximal Spectral Type of Unitary Operators Adjoint to Gauss Dynamical Systems.- x4. Gauss Dynamical Systems with Simple Continuous Spectrum.- x5. Gauss Dynamical Systems with Finite Multiplicity Spectrum.- IV Approximation Theory of Dynamical Systems by Periodic Dynamical Systems and Some of its Applications.- 15 Approximations of Dynamical Systems.- x1. Definition and Types of Approximations. Ergodicity and Mixing Conditions.- x2. Approximations and Spectra.- x3. An Application of Approximation Theory: an Example of an Ergodic Automorphism with a Spectrum Lacking the Group Property.- x4. Approximation of Flows.- 16 Special Representations and Approximations of Smooth Dynamical Systems on the Two-dimensional Torus.- x1. Special Representations of Flows on the Torus.- x2. Dynamical Systems with Pure Point Spectrum on the Two-dimensional Torus.- x3. Approximations of Flows on the Torus.- x4. Example of a Smooth Flow with Continuous Spectrum on the Two-dimensional Torus.- Appendix 1.- Lebesgue Spaces and Measurable Partitions.- Appendix 2.- Relevant Facts from the Spectral Theory of Unitary Operators.- Appendix 3.- Proof of the Birkhoff-Khinchin Theorem.- Appendix 4.- Kronecker Sets.- Bibliographical Notes.