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図書
図書
1. Dynamical systems, ergodic theory and applications. 2nd, expanded and rev. ed
L.A. Bunimovich ... [et al.] ; edited by Ya.G. Sinai
出版情報: Berlin ; Tokyo : Springer-Verlag, c2000  x, 459 p. ; 24 cm
シリーズ名: Encyclopaedia of mathematical sciences / editor-in-chief, R.V. Gamkrelidze ; v. 100 . Mathematical physics ; 1
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目次情報: 続きを見る
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
General Ergodic Theory of Groupsof Measure Preserving Transformations / I：
Ergodic Theory of Smooth Dynamical Systems / II：
Dynamical Systems on Homogeneous Spaces / III：
2.

図書
図書
2. Hard ball systems and the Lorentz gas
L.A. Bunimovich ... [et al.] ; edited by D. Szász
出版情報: Berlin ; Tokyo : Springer-Verlag, c2000  viii, 458 p. ; 24 cm
シリーズ名: Encyclopaedia of mathematical sciences / editor-in-chief, R.V. Gamkrelidze ; v. 101 . Mathematical physics ; 2
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目次情報: 続きを見る
Mathematics / Part I：
A Geometric Approach to Semi-Dispersing Billiards / D. Burago ; S. Ferleger ; A. Kononenko1：
On the Sequences of Collisions Among Hard Spheres in Infinite Spacel / T. J. Murphy ; E. G. D. Cohen2：
Hard Ball Systems and Semi-Dispersive Billiards: Hyperbolicity and Ergodicity / N. Simanyi3：
Decay of Correlations for Lorentz Gases and Hard Balls / N. Chernov ; L.-S Young4：
Entropy Values and Entropy Bounds / 5：
Existence of Transport Coefficients / L. A. Bunimovich6：
Interacting Particles / C. Liverani7：
Scaling Dynamics of a Massive Piston in an Ideal Gas / J. L. Lebowitz ; J. Piasecki ; Ya G. Sinai8：
Physics / Part II：
Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy, and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases / H. van Beijeren ; R. van Zon ; J. R. Dorfman
Simulation of Billiards and of Hard-Body Fluids / H. A. Posch ; R. Hirschl
The Lorentz Gas: a Paradigm for Nonequilibrium Stationary States / C. P. Dettmann
T. Tl, J. Vollmer: Entropy Balance, Multibaker Maps, and the Dynamics of the Lorentz Gas
Boltzmanns Ergodic Hypothesis, a Conjecture for Centuries? / D. SzaszAppendix 1：
Mathematics / Part I：
A Geometric Approach to Semi-Dispersing Billiards / D. Burago ; S. Ferleger ; A. Kononenko1：
On the Sequences of Collisions Among Hard Spheres in Infinite Spacel / T. J. Murphy ; E. G. D. Cohen2：
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