Ergodic Theory.

Format:

Book

Author/Creator:

Rubinstein-Salzedo, Simon.

Series:

The Carus Mathematical Monographs

The Carus Mathematical Monographs ; v.38

Language:

English

Subjects (All):

Ergodic theory--Textbooks.

Ergodic theory.

Dynamics--Textbooks.

Dynamics.

Number theory--Textbooks.

Number theory.

Physical Description:

1 online resource (227 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2025.

Summary:

Ergodic theory is concerned with the measure-theoretic or statistical properties of a dynamical system. This book provides a conversational introduction to the topic, guiding the reader from the classical questions of measure theory to modern results such as the polynomial recurrence theorem. Applications to number theory and combinatorics enhance the exposition, while also presenting the utility of ergodic theory in other areas of research. The book begins with an introduction to measure theory and the Lebesgue integral. After this, the key concepts of the subject are covered: measure-preserving transformations, ergodicity, and invariant measures. These chapters also cover classical results such as Poincaré's recurrence theorem and Birkhoff's ergodic theorem. The book ends with more advanced topics, such as mixing, entropy, and an appendix on the weak* topology. Each chapter ends with numerous exercises with a range of difficulty levels, including a handful of open problems. An excellent resource for anyone wishing to learn about ergodic theory, the book only assumes prior exposure to proof-based mathematics. Familiarity with real analysis would be ideal but is not required.

Contents:

Intro

Title page

Copyright

Contents

Preface

Introduction

0.1. What is Ergodic Theory?

0.2. Motivating Examples

0.3. The Power of Ergodic Theory

0.4. Structure of the Book

0.5. Prerequisites

Chapter 1. Introduction to Measure Theory

1.1. Measurable Difficulties

1.2. Measure Spaces

1.3. The Borel -algebra

1.4. Outer Measures

1.5. Measures from Outer Measures

1.6. Problems

Chapter 2. The Lebesgue Integral

2.1. Lebesgue Sets

2.2. The Riemann Integral

2.3. The Lebesgue Integral

2.4. New Measures from Integration

2.5. Problems

Chapter 3. Some Limit Theorems

3.1. Riemann Integrals Misbehaving

3.2. The Big Three

3.3. An Application

3.4. Types of Convergence

3.5. ^{ } Spaces

3.6. Problems

Chapter 4. Measure-Preserving Transformations

4.1. Measure-Preserving Transformations

4.2. Digital Expansions

4.3. Continued Fractions

4.4. Lüroth Series

4.5. Problems

Chapter 5. The Poincaré* Recurrence Theorem

5.1. Recurrence

5.2. The Poincaré Recurrence Theorem

5.3. A First Look at Multiple Recurrence

5.4. Invariant Sets and Ergodicity

5.5. Induced Transformations

5.6. Problems

Chapter 6. Ergodicity

6.1. Proving Ergodicity

6.2. The Strong Law of Large Numbers

6.3. The Birkhoff Ergodic Theorem

6.4. Proof of the Ergodic Theorem

6.5. Problems

Chapter 7. Invariant Measures

7.1. Continued Fractions

7.2. Existence of Invariant Measures

7.3. Ergodicity of Invariant Measures

7.4. Unique Ergodicity

7.5. Problems

Chapter 8. Mixing

8.1. Mixing

8.2. Types of Convergence

8.3. Weak Mixing

8.4. Weakly Mixing Does Not Imply Mixing

8.5. Problems

Chapter 9. Multiple Recurrence* and Szemerédi's Theorem

9.1. A Brief Look at Additive Combinatorics

9.2. The Furstenberg Multiple Recurrence * Theorem.

9.3. The Furstenberg Correspondence* Principle

9.4. Arbitrarily Long Arithmetic Progressions

9.5. Schur's Work on Arithmetic Progressions

9.6. Problems

Chapter 10. Polynomial Recurrence

10.1. Polynomial Recurrence

10.2. Chaotic and Orderly Parts

10.3. mcH _{ at }

10.4. mcV _{∗}

10.5. Problems

Chapter 11. Entropy

11.1. Introduction to Entropy

11.2. Computing Entropy with the* Kolmogorov-SinaiTheorem

11.3. Conditional Expectation with* Measure Theory

11.4. The Shannon-McMillan-Breiman* Theorem

11.5. Lochs's Theorem

11.6. Problems

Appendix A. The Weak* Topology

A.1. Topological Spaces

A.2. Continuity

A.3. The Weak* Topology

A.4. The Space of Borel Probability Measures

Bibliography

Index.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-8048-4