Basic ergodic theory
著者
書誌事項
Basic ergodic theory
（Birkhäuser advanced texts : Basler Lehrbücher / edited by Herbert Amann, Hanspeter Kraft）
Birkhäuser Verlag, c1995
2nd ed
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注記
Includes bibliographical references (p. [141]-146) and index
内容説明・目次
内容説明
This is an introductory book on ergodic theory. The presentation has a slow pace and the book can be read by anyone with a background in basic measure theory and metric topology. In particular, the first two chapters, the elements of ergodic theory, can form a course of four to six lectures at the advanced undergraduate or the beginning graduate level. A new feature of the book is that the basic topics of ergodic theory such as the Poincare recurrence lemma, induced automorphisms and Kakutani towers, compressibility and E. Hopf's theorem, the theorem of Ambrose on representation of flows are treated at the descriptive set-theoretic level before their measure-theoretic or topological versions are presented. In addition, topics centering around the Glimm-Effros theorem are discussed, topics which have so far not found a place in texts on ergodic theory. In this second edition, a section on rank one automorphisms and a brief discussion of the ergodic theorem due to Wiener and Wintner have been added. "This relatively short book is, for anyone new to ergodic theory, admirably broad in scope.
The exposition is clear, and the brevity of the book has not been achieved by giving terse proofs. The examples have been chosen with great care. Historical facts and many references serve to help connect the reader with literature that goes beyond the content of the book as well as explaining how the subject developed. It is easy to recommend this book for students as well as anyone who would like to learn about the descriptive approach to ergodic theory." (Summary of a review of the first edition in Math Reviews)
目次
- The Poincare recurrence lemma - standard Borel spaces, Borel automorphisms, orbit equivalence and isomorphism, poincare recurrence lemma, asides
- ergodic theorems of Birkhoff and von Neumann - ergodic theorem for permutations, easy generalisations, almost periodic functions, Birkhoff's ergodic theorem, von Neumann ergodic theorem, ergodic theorems for return times, asides
- ergodicity - discussion of ergodicity, irrational rotation, diadic adding machine
- mixing conditions and their characterisations
- Bernoulli shift and related concepts - Bernoulli shifts, Kolmogorov consistency theorem, Markov shift, Kolmogorov shifts and related concepts, non-invertible shifts, diadic adding machine, Hewitt-Savage Zero-one law
- discrete spectrum theorem - spectral isomorphism of Brenoulli shifts, entropy
- induced automorphisms and related concepts - Kakutani towers, periodic approximations, Rokhlin's lemma, induced automorphisms, automorphism built under a function, Kakutani equivalence, rank of an automorphism, countable generators, rank of an automorphism
- Borel automorphisms are polish homeomorphisms
- the Glimm-Effros theorem
- E. Hopf's theorem -compressibility (in the sense of Brikhoff and Smith), compressibility (in the sense of Hopf), compressibility, ergodic theorem (measure free proof), ergodic decomposition, proof of Hopf's theorem, orbit equivalence, generalisations and counter-examples
- H. Dye's theorem
- flows and their representations - definitions and examples, flow built under a function, topology for a flow, existence of countable cross-sections, representation of non-singular flows, representation of measure preserving flows.
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