Dynamical Systems, Ergodic Theory and Applications

Transcription

1 Encyclopaedia of Mathematical Sciences 100 Dynamical Systems, Ergodic Theory and Applications Bearbeitet von L.A. Bunimovich, S.G. Dani, R.L. Dobrushin, M.V. Jakobson, I.P. Kornfeld, N.B. Maslova, Ya.B. Pesin, Ya.G. Sinai, J. Smillie, Yu.M. Sukhov, A.M. Vershik erweitert, überarbeitet Buch. xi, 460 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 1850 g Weitere Fachgebiete > Mathematik > Geometrie schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.

2 Contents I. General Ergodic Theory of Groups of Measure Preserving Transformations 1 II. Ergodic Theory of Smooth Dynamical Systems 103 III. Dynamical Systems on Homogeneous Spaces 264 IV. The Dynamics of Billiard Flows in Rational Polygons 360 V. Dynamical Systems of Statistical Mechanics and Kinetic Equations 383 Subject Index 455

3 I. General Ergodic Theory of Groups of Measure Preserving Transformations Contents Chapter 1. Basic Notions of Ergodic Theory and Examples of Dynamical Systems (I.P. Kornfeld, Ya.G. Sinai) Dynamical Systems with Invariant Measures First Corollaries of the Existence of Invariant Measures. Ergodic Theorems Ergodicity. Decomposition into Ergodic Components. Various Mixing Conditions General Constructions Direct Products of Dynamical Systems Skew Products of Dynamical Systems Factor-Systems Integral and Induced Automorphisms Special Flows and Special Representations of Flows Natural Extensions of Endomorphisms Chapter 2. Spectral Theory of Dynamical Systems (I.P. Kornfeld, Ya.G. Sinai) 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 Chapter 3. Entropy Theory of Dynamical Systems (I.P. Kornfeld, Ya.G. Sinai) Entropy and Conditional Entropy of a Partition Entropy of a Dynamical System The Structure of Dynamical Systems of Positive Entropy... 43

4 Contents The Isomorphy Problem for Bernoulli Automorphisms and K-Systems Equivalence of Dynamical Systems in the Sense of Kakutani 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 (I.P. Kornfeld, A.M. Vershik) 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 General Definition of the Actions of Locally Compact Groups on Lebesgue Spaces Ergodic Theorems Spectral Theory Entropy Theory for the Actions of General Groups Chapter 5. Trajectory Theory (A.M. Vershik) 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

5 II. Ergodic Theory of Smooth Dynamical Systems Contents Chapter 6. Stochasticity of Smooth Dynamical Systems. The Elements of KAM-Theory (Ya.G. Sinai) Integrable and Nonintegrable Smooth Dynamical Systems. The Hierarchy of Stochastic Properties of Deterministic Dynamics The Kolmogorov-Arnold-Moser Theory (KAM-Theory) Chapter 7. General Theory of Smooth Hyperbolic Dynamical Systems (Ya.B. Pesin) Hyperbolicity of Individual Trajectories Introductory Remarks Uniform Hyperbolicity Nonuniform Hyperbolicity Local Manifolds Global Manifolds Basic Classes of Smooth Hyperbolic Dynamical Systems. Definitions and Examples Anosov Systems Hyperbolic Sets Locally Maximal Hyperbolic Sets Axiom A-Diffeomorphisms Hyperbolic Attractors. Repellers Partially Hyperbolic Dynamical Systems Mather Theory Nonuniformely Hyperbolic Dynamical Systems. Lyapunov Exponents Ergodic Properties of Smooth Hyperbolic Dynamical Systems u-gibbs Measures Symbolic Dynamics Measures of Maximal Entropy Construction of u-gibbs Measures

6 104 Contents 3.5. Topological Pressure and Topological Entropy Properties of u-gibbs Measures Small Stochastic Perturbations Equilibrium States and Their Ergodic Properties Ergodic Properties of Dynamical Systems with Nonzero Lyapunov Exponents Ergodic Properties of Anosov Systems and of UPH-Systems Continuous Time Dynamical Systems 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 Introductory Remarks Hausdorff Dimension Other Dimension Characteristics Carathéodory Dimension Structure. Carathéodory Dimension Characteristics Examples of C-structures and Carathéodory Dimension Characteristics Multifractal Formalism Coupled Map Lattices Additional References Chapter 8. Billiards and Other Hyperbolic Systems (L.A. Bunimovich) 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 Billiards in Domains with Boundary Possessing Focusing Components Hyperbolic Dynamical Systems with Singularities (a General Approach) Markov Approximations and Symbolic Dynamics for Hyperbolic Billiards Statistical Properties of Dispersing Billiards and of the Lorentz Gas Transport Coefficients for the Simplest Mechanical Models 222

7 Contents Strange Attractors Definition of a Strange Attractor The Lorenz Attractor Some Other Examples of Hyperbolic Strange Attractors Additional References Chapter 9. Ergodic Theory of One-Dimensional Mappings (M.V. Jakobson) 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 Bibliography

8 III. Dynamical Systems on Homogeneous Spaces Contents Chapter 10. Dynamical Systems on Homogeneous Spaces (S.G. Dani) Introduction Measures on homogeneous spaces 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 Actions of groups of affine automorphisms 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

9 Contents Mixing of higher orders Entropy K-mixing, Bernoullicity Group-induced flows with hyperbolic structure Anosov automorphisms Affine automorphisms with a hyperbolic fixed point Anosov flows 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 Orbit closures of group-induced flows Homogeneity of orbit closures Orbit closures of horospherical subgroups Orbits of reductive subgroups Orbit closures of one-parameter flows Dense orbits and minimal sets of flows Divergent trajectories of flows Bounded orbits and escapable sets Duality and lattice-actions on vector spaces Duality between orbits Duality of invariant measures Applications to Diophantine approximation Polynomials in one variable Values of linear forms Diophantine approximation with dependent quantities Values of quadratic forms Forms of higher degree Integral points on algebraic varieties Classification and related questions Metric isomorphisms and factors Metric rigidity Topological conjugacy Topological equivalence Bibliography

10 IV. The Dynamics of Billiard Flows in Rational Polygons Contents Chapter 11. The Dynamics of Billiard Flows in Rational Polygons of Dynamical Systems (J. Smillie) 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 Bibliography

11 V. Dynamical Systems of Statistical Mechanics and Kinetic Equations Contents Chapter 12. Dynamical Systems of Statistical Mechanics (R.L. Dobrushin, Ya.G. Sinai, Yu.M. Sukhov) Introduction 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 P t Ideal Gas and Related Systems The Poisson Superstructure Asymptotic Behaviour of the Probability Distribution P t as t The Dynamical System of One-Dimensional Hard Rods Kinetic Equations Statement of the Problem The Boltzmann Equation The Vlasov Equation The Landau Equation Hydrodynamic Equations Bibliography

12 Chapter 13. Existence and Uniqueness Theorems for the Boltzmann Equation (N.B. Maslova) Formulation of Boundary Problems. Properties of Integral Operators The Boltzmann Equation 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 Relaxation in a Homogeneous Gas The Cauchy Problem Boundary Problems On a Connection of the Boltzmann Equation with Hydrodynamic Equations Statement of the Problem Local Solutions. Reduction to Euler Equations A Global Theorem. Reduction to Navier-Stokes Equations. 450 Bibliography