遍歷性理論引論(北京)

遍歷性理論引論(北京)
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內容簡介

In 1970 I gave a graduate course in ergodic theory at the University of Maryland in College Park, and these lectures were the basis of the Springer Lecture Notes in Mathematics Volume 458 called ”Ergodic Theory--Introductory Lectures” which was published in 1975. This volume is nowout of print, so I decided to revise and add to the contents of these notes. I have updated the earlier chapters and have added some new chapters on the ergodic theory of continuous transformations of compact metric spaces. In particular, I have included some material on topological pressure and equilibrium states. In recent years there have been some fascinating interactions of ergodic theory with differentiable dynamics, differential geometry,number theory, von Neumann algebras, probability theory, statistical mechanics, and other topics. In Chapter 10 1 have briefly described some of these and given references to some of the others. I hope that this book will give the reader enough foundation to tackle the research papers on ergodictheory and its applications.
 

目錄

Chapter 0 Preliminaries
　0.1 Introduction
　0.2 Measure Spaces
　0.3 Integration
　0.4 Absolutely Continuous Measures and Conditional Expectations
　0.5 Function Spaces
　0.6 Haar Measure
　0.7 Character Theory
　0.8 Endomorphisms of Tori
　0.9 Perron-Frobenius Theory
　0.10 Topology

Chapter 1 Measure-Preserving Transformations
　1.1 Definition and Examples
　1.2 Problems in Ergodic Theory
　1.3 Associated Isometries
　1.4 Recurrence
　1.5 Ergodicity
　1.6 The Ergodic Theorem
　1.7 Mixing

Chapter 2 Isomorphism, Conjugacy, and Spectral Isomorphism
　2.1 Point Maps and Set Maps
　2.2 Isomorphism of Measure-Preserving Transformations
　2.3 Conjugacy of Measure-preserving Transformhtions
　2.4 The Isomorphism Problem
　2.5 Spectral Isomorphism
　2.6 Spectral Invariants

Chapter 3 Measure-Preserving Transformations with Discrete Spectrum
　3.1 Eigenvalues and Eigenfunctions
　3.2 Discrete Spectrum
　3.3 Group Rotations

Chapter 4 Entropy
　4.1 Partitions and Subalgebras
　4.2 Entropy of a Partition
　4.3 Conditional Entropy
　4.4 Entropy of a Measure-Preserving Transformation
　4.5 Properties orb T,A and h T
　4.6 Some Methods for Calculating h T
　4.7 Examples
　4.8 How Good an Invariant is Entropy
　4.9 Bernoulli Automorphisms and Kolmogorov Automorphisms
　4.10 The Pinsker -Algebra of a Measure-Preserving Transformation
　4.11 Sequence Entropy
　4.12 Non-invertible Transformations
　4.13 Comments

Chapter 5 Topological Dynamics
　5.1 Examples
　5.2 Minimality
　5.3 The Non-wandering Set
　5.4 Topological Transitivity
　5.5 Topological Conjugacy and Discrete Spectrum
　5.6 Expansive Homeomorphisms

Chapter 6 Invariant Measures for Continuous Transformations
　6.1 Measures on Metric Spaces
　6.2 Invariant Measures for Continuous Transformations
　6.3 Interpretation of Ergodicity and Mixing
　6.4 Relation of Invariant Measures to Non-wandering Sets, Periodic Points and Topological Transitivity
6.5 Unique Ergodicity
6.6 Examples

Chapter 7 Topological Entropy
　7.1 Definition Using Open Covers
　7.2 Bowen’’s Definition
　7.3 Calculation of Topological Entropy

Chapter 8 Relationship Between Topological Entropy and Measure-Theoretic Entropy
　8.1 The Entropy Map
　8.2 The Variational Principle
　8.3 Measures with Maximal Entropy
　8.4 Entropy of Affine Transformations
　8.5 The Distribution of Periodic Points
　8.6 Definition of Measure-Theoretic Entropy Using the Metrics dn

Chapter 9 Topological Pressure and Its Relationship with Invariant Measures
　9.1 Topological Pressure
　9.2 Properties of Pressure
　9.3 The Variational Principle
　9.4 Pressure Determines M X, T
　9.5 Equilibrium States

Chapter 10 Applications and Other Topics
　10.1 The Qualitative Behaviour of Diffeomorphisms
　10.2 The Subadditive Ergodic Theorem and the Multiplicative Ergodic Theorem
　10.3 Quasi-invariant Measures
　10.4 Other Types of Isomorphism
　10.5 Transformations of Intervals
　10.6 Further Reading

References
Index
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