Ebook: Basic Ergodic Theory

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 Poincar? 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)
Content:
Front Matter....Pages i-ix
The Poincar? Recurrence Lemma....Pages 1-11
Ergodic Theorems of Birkhoff and von Neumann....Pages 13-32
Ergodicity....Pages 33-42
Mixing Conditions and Their Characterisations....Pages 43-50
Bernoulli Shift and Related Concepts....Pages 51-59
Discrete Spectrum Theorem....Pages 61-64
Induced Automorphisms and Related Concepts....Pages 65-77
Borel Automorphisms are Polish Homeomorphisms....Pages 79-82
The Glimm-Effros Theorem....Pages 83-92
E. Hopf’s Theorem....Pages 93-112
H. Dye’s Theorem....Pages 113-124
Flows and Their Representations....Pages 125-139
Back Matter....Pages 141-150

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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 Poincar? 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)
Content:
Front Matter....Pages i-ix
The Poincar? Recurrence Lemma....Pages 1-11
Ergodic Theorems of Birkhoff and von Neumann....Pages 13-32
Ergodicity....Pages 33-42
Mixing Conditions and Their Characterisations....Pages 43-50
Bernoulli Shift and Related Concepts....Pages 51-59
Discrete Spectrum Theorem....Pages 61-64
Induced Automorphisms and Related Concepts....Pages 65-77
Borel Automorphisms are Polish Homeomorphisms....Pages 79-82
The Glimm-Effros Theorem....Pages 83-92
E. Hopf’s Theorem....Pages 93-112
H. Dye’s Theorem....Pages 113-124
Flows and Their Representations....Pages 125-139
Back Matter....Pages 141-150
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