General Ergodic Theory of Groupsof Measure Preserving Transformations / I: |
Ergodic Theory of Smooth Dynamical Systems / II: |
Dynamical Systems on Homogeneous Spaces / III: |
The Dynamics of Billiard Flowsin Rational Polygons / IV: |
Dynamical Systemsof Statistical Mechanics and Kinetic Equations / V: |
Subject Index |
Basic Notions of Ergodic Theory and Examples of Dynamical Systems / I.P. Kornfeld ; Ya.G. SinaiChapter 1: |
Dynamical Systems with Invariant Measures / 1: |
First Corollaries of the Existence of Invariant Measures. Ergodic Theorems / 2: |
Ergodicity. Decomposition into Ergodic Components. Various Mixing Conditions / 3: |
General Constructions / 4: |
Direct Products of Dynamical Systems / 4.1: |
Skew Products of Dynamical Systems / 4.2: |
Factor-Systems / 4.3: |
Integral and Induced Automorphisms / 4.4: |
Special Flows and Special Representations of Flows / 4.5: |
Natural Extensions of Endomorphisms / 4.6: |
Spectral Theory of Dynamical Systems / Chapter 2: |
Groups of Unitary Operators and Semigroups of Isometric Operators Adjoint to Dynamical Systems |
The Structure of the Dynamical Systems with Pure Point and Quasidiscrete Spectra |
Examples of Spectral Analysis of Dynamical Systems |
Spectral Analysis of Gauss Dynamical Systems |
Entropy Theory of Dynamical Systems / Chapter 3: |
Entropy and Conditional Entropy of a Partition |
Entropy of a Dynamical System |
The Structure of Dynamical Systems of Positive Entropy |
The Isomorphy Problem for Bernoulli Automorphisms and K-Systems |
Equivalence of Dynamical Systems in the Sense of Kakutani / 5: |
Shifts in the Spaces of Sequences and Gibbs Measures / 6: |
Periodic Approximations and Their Applications. Ergodic Theorems, Spectral and Entropy Theory for the General Group Actions / A.M. VershikChapter 4: |
Approximation Theory of Dynamical Systems by Periodic Ones. Flows on the Two-Dimensional Torus |
Flows on the Surfaces of Genus p ≥ 1 and Interval Exchange Transformations |
General Group Actions |
Introduction / 3.1: |
General Definition of the Actions of Locally Compact Groups on Lebesgue Spaces / 3.2: |
Ergodic Theorems / 3.3: |
Spectral Theory / 3.4: |
Entropy Theory for the Actions of General Groups |
Trajectory Theory / Chapter 5: |
Statements of Main Results |
Sketch of the Proof. Tame Partitions |
Trajectory Theory for Amenable Groups |
Trajectory Theory for Non-Amenable Groups. Rigidity |
Concluding Remarks. Relationship Between Trajectory Theory and Operator Algebras |
Bibliography |
Additional Bibliography |
Ergodic Theory of SmoothDynamical Systems |
Stochasticity of Smooth Dynamical Systems. The Elements of KAM-Theory / Chapter 6: |
Integrable and Nonintegrable Smooth Dynamical Systems. The Hierarchy of Stochastic Properties of Deterministic Dynamics |
The Kolmogorov-Arnold-Moser Theory (KAM-Theory) |
General Theory of Smooth Hyperbolic Dynamical Systems / Ya.B.PesinChapter 7: |
Hyperbolicity of Individual Trajectories |
Introductory Remarks / 1.1: |
Uniform Hyperbolicity / 1.2: |
Nonuniform Hyperbolicity / 1.3: |
Local Manifolds / 1.4: |
Global Manifolds / 1.5: |
Basic Classes of Smooth Hyperbolic Dynamical Systems. Definitions and Examples |
Anosov Systems / 2.1: |
Hyperbolic Sets / 2.2: |
Locally Maximal Hyperbolic Sets / 2.3: |
Axiom A-Diffeomorphisms / 2.4: |
Hyperbolic Attractors. Repellers / 2.5: |
Partially Hyperbolic Dynamical Systems / 2.6: |
Mather Theory / 2.7: |
Nonuniformely Hyperbolic Dynamical Systems. Lyapunov Exponents / 2.8: |
Ergodic Properties of Smooth Hyperbolic Dynamical Systems |
u-Gibbs Measures |
Symbolic Dynamics |
Measures of Maximal Entropy |
Construction of u-Gibbs Measures |
Topological Pressure and Topological Entropy / 3.5: |
Properties of u-Gibbs Measures / 3.6: |
Small Stochastic Perturbations / 3.7: |
Equilibrium States and Their Ergodic Properties / 3.8: |
Ergodic Properties of Dynamical Systems with Nonzero Lyapunov Exponents / 3.9: |
Ergodic Properties of Anosov Systems and of UPH-Systems / 3.10: |
Continuous Time Dynamical Systems / 3.11: |
Hyperbolic Geodesic Flows |
Manifolds with Negative Curvature |
Riemannian Metrics Without Conjugate (or Focal) Points |
Entropy of Geodesic Flows |
Riemannian Metrics of Nonpositive Curvature |
Geodesic Flows on Manifolds with Constant Negative Curvature |
Dimension-like Characteristics of Invariant Sets for Dynamical Systems |
Hausdorff Dimension / 6.1: |
Other Dimension Characteristics / 6.3: |
Carathéodory Dimension Structure. Carathéodory Dimension Characteristics / 6.4: |
Examples of C-structures and Carathéodory Dimension Characteristics / 6.5: |
Multifractal Formalism / 6.6: |
Coupled Map Lattices / 7: |
Additional References |
Billiards and Other Hyperbolic Systems / L.A. BunimovichChapter 8: |
Billiards |
The General Definition of a Billiard |
Billiards in Polygons and Polyhedrons |
Billiards in Domains with Smooth Convex Boundary |
Dispersing or Sinai Billiards |
The Lorentz Gas and Hard Spheres Gas |
Semi-dispersing Billiards and Boltzmann Hypotheses / 1.6: |
Billiards in Domains with Boundary Possessing Focusing Components / 1.7: |
Hyperbolic Dynamical Systems with Singularities (a General Approach) / 1.8: |
Markov Approximations and Symbolic Dynamics for Hyperbolic Billiards / 1.9: |
Statistical Properties of Dispersing Billiards and of the Lorentz Gas / 1.10: |
Transport Coefficients for the Simplest Mechanical Models / 1.11: |
Strange Attractors |
Definition of a Strange Attractor |
The Lorenz Attractor |
Some Other Examples of Hyperbolic Strange Attractors |
Ergodic Theory of One-Dimensional Mappings / M.V. JakobsonChapter 9: |
Expanding Maps |
Definitions, Examples, the Entropy Formula |
Walters Theorem |
Absolutely Continuous Invariant Measures for Nonexpanding Maps |
Some Examples |
Intermittency of Stochastic and Stable Systems |
Ergodic Properties of Absolutely Continuous Invariant Measures |
Feigenbaum Universality Law |
The Phenomenon of Universality |
Doubling Transformation |
Neighborhood of the Fixed Point |
Properties of Maps Belonging to the Stable Manifold of Φ |
Rational Endomorphisms of the Riemann Sphere |
The Julia Set and Its Complement |
The Stability Properties of Rational Endomorphisms |
Ergodic and Dimensional Properties of Julia Sets |
Dynamical Systemson Homogeneous Spaces |
Measures on homogeneous spaces / S.G. DaniChapter 10: |
Examples of lattices |
Ergodicity and its consequences |
Isomorphisms and factors of affine automorphisms |
Affine automorphisms of tori and nilmanifolds |
Ergodic properties; the case of tori |
Ergodic properties on nilmanifolds |
Unipotent affine automorphisms |
Quasi-unipotent affine automorphisms |
Closed invariant sets of automorphisms |
Dynamics of hyperbolic automorphisms |
More on invariant sets of hyperbolic toral automorphisms |
Distribution of orbits of hyperbolic automorphisms |
Dynamics of ergodic toral automorphisms / 2.9: |
Actions of groups of affine automorphisms / 2.10: |
Group-induced translation flows; special cases |
Flows on solvmanifolds |
Homogeneous spaces of semisimple groups |
Flows on low-dimensional homogeneous spaces |
Ergodic properties of flows on general homogeneous spaces |
Horospherical subgroups and Mautner phenomenon |
Ergodicity of one-parameter flows |
Invariant functions and ergodic decomposition |
Actions of subgroups |
Duality |
Spectrum and mixing of group-induced flows |
Mixing of higher orders / 4.7: |
Entropy / 4.8: |
K-mixing, Bernoullicity / 4.9: |
Group-induced flows with hyperbolic structure |
Anosov automorphisms / 5.1: |
Affine automorphisms with a hyperbolic fixed point / 5.2: |
Anosov flows / 5.3: |
Invariant measures of group-induced flows |
Invariant measures of Ad-unipotent flows |
Invariant measures and epimorphic subgroups |
Invariant measures of actions of diagonalisable groups |
A weak recurrence property and infinite invariant measures |
Distribution of orbits and polynomial trajectories |
A uniform version of uniform distribution |
Distribution of translates of closed orbits / 6.7: |
Orbit closures of group-induced flows |
Homogeneity of orbit closures / 7.1: |
Orbit closures of horospherical subgroups / 7.2: |
Orbits of reductive subgroups / 7.3: |
Orbit closures of one-parameter flows / 7.4: |
Dense orbits and minimal sets of flows / 7.5: |
Divergent trajectories of flows / 7.6: |
Bounded orbits and escapable sets / 7.7: |
Duality and lattice-actions on vector spaces / 8: |
Duality between orbits / 8.1: |
Duality of invariant measures / 8.2: |
Applications to Diophantine approximation / 9: |
Polynomials in one variable / 9.1: |
Values of linear forms / 9.2: |
Diophantine approximation with dependent quantities / 9.3: |
Values of quadratic forms / 9.4: |
Forms of higher degree / 9.5: |
Integral points on algebraic varieties / 9.6: |
Classification and related questions / 10: |
Metric isomorphisms and factors / 10.1: |
Metric rigidity / 10.2: |
Topological conjugacy / 10.3: |
Topological equivalence / 10.4: |
The Dynamics of Billiard Flows in Rational Polygons of Dynamical Systems / J. SmillieChapter 11: |
Two Examples |
Formal Properties of the Billiard Flow |
The Flow in a Fixed Direction |
Billiard Techniques: Minimality and Closed Orbits |
Billiard Techniques: Unique Ergodicity |
Dynamics on Moduli Spaces |
The Lattice Examples of Veech |
Dynamical Systems of Statistical Mechanicsand Kinetic Equations |
Dynamical Systems of Statistical Mechanics / R.L. Dobrushin ; Yu.M. SukhovChapter 12: |
Phase Space of Systems of Statistical Mechanics and Gibbs Measures |
The Configuration Space |
Poisson Measures |
The Gibbs Configuration Probability Distribution |
Potential of the Pair Interaction. Existence and Uniqueness of a Gibbs Configuration Probability Distribution |
The Phase Space. The Gibbs Probability Distribution |
Gibbs Measures with a General Potential |
The Moment Measure and Moment Function |
Dynamics of a System of Interacting Particles |
Statement of the Problem |
Construction of the Dynamics and Time Evolution |
Hierarchy of the Bogolyubov Equations |
Equilibrium Dynamics |
Definition and Construction of Equilibrium Dynamics |
The Gibbs Postulate |
Degenerate Models |
Asymptotic Properties of the Measures Pt |
Ideal Gas and Related Systems |
The Poisson Superstructure |
Asymptotic Behaviour of the Probability Distribution Pt as t → ∞ |
The Dynamical System of One-Dimensional Hard Rods |
Kinetic Equations |
The Boltzmann Equation |
The Vlasov Equation |
The Landau Equation |
Hydrodynamic Equations |
Existence and Uniqueness Theorems for the Boltzmann Equation / N.B. MaslovaChapter 13: |
Formulation of Boundary Problems. Properties of Integral Operators |
Formulation of Boundary Problems |
Properties of the Collision Integral |
Linear Stationary Problems |
Asymptotics |
Internal Problems |
External Problems |
Kramers' Problem |
Nonlinear Stationary Problems |
Non-Stationary Problems |