Ergodic Theory. Questo articolo non è disponibile.Ergodic Theory

Ergodic theory is one of the few branches of mathematics which has changed radically during the last two decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily with averaging problems and general qualitative questions, while now it is a powerful amalgam of methods used for the analysis of statistical properties of dyna­ mical systems. For this reason, the problems of ergodic theory now interest not only the mathematician, but also the research worker in physics, biology, chemistry, etc. The outline of this book became clear to us nearly ten years ago but, for various reasons, its writing demanded a long period of time. The main principle, which we adhered to from the beginning, was to develop the approaches and methods or ergodic theory in the study of numerous concrete examples. Because of this, Part I of the book contains the description of various classes of dynamical systems, and their elementary analysis on the basis of the fundamental notions of ergodicity, mixing, and spectra of dynamical systems. Here, as in many other cases, the adjective" elementary" i~ not synonymous with "simple. " Part II is devoted to "abstract ergodic theory. " It includes the construc­ tion of direct and skew products of dynamical systems, the Rohlin-Halmos lemma, and the theory of special representations of dynamical systems with continuous time. A considerable part deals with entropy.

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