**Student Mathematical Library**

Volume: 42;
2018;
262 pp;
Softcover

MSC: Primary 37; 28; 54;

Print ISBN: 978-0-8218-4420-5

Product Code: STML/42

List Price: $49.00

Individual Price: $39.20

**Electronic ISBN: 978-1-4704-2151-9
Product Code: STML/42.E**

List Price: $49.00

Individual Price: $39.20

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#### Supplemental Materials

# Invitation to Ergodic Theory

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*C. E. Silva*

This book is an introduction to basic concepts in ergodic theory such
as recurrence, ergodicity, the ergodic theorem, mixing, and weak
mixing. It does not assume knowledge of measure theory; all the
results needed from measure theory are presented from scratch. In
particular, the book includes a detailed construction of the Lebesgue
measure on the real line and an introduction to measure spaces up to
the Carathéodory extension theorem. It also develops the
Lebesgue theory of integration, including the dominated convergence
theorem and an introduction to the Lebesgue \(L^p\)spaces.

Several examples of a dynamical system are developed in detail to
illustrate various dynamical concepts. These include in particular the
baker's transformation, irrational rotations, the dyadic odometer, the
Hajian–Kakutani transformation, the Gauss transformation, and
the Chacón transformation. There is a detailed discussion of
cutting and stacking transformations in ergodic theory. The book
includes several exercises and some open questions to give the flavor
of current research. The book also introduces some notions from
topological dynamics, such as minimality, transitivity and symbolic
spaces; and develops some metric topology, including the Baire
category theorem.

#### Readership

Undergraduate and graduate students interested in ergodic theory and measure theory.

#### Reviews & Endorsements

...comprehensive in scope, uncovering key ideas ranging from Euclidean geometry to transformations to affine systems to non-Euclidean geometries. ...The authors neither cut corners nor 'wave' at neat ideas; rather, they try to connect everything via a quite rigorous development, complete with well-chosen exercises.

-- CHOICE Reviews

The writing is crisp and clear. Proofs are written carefully with adequate levels of detail. Exercises are plentiful and well-integrated with the text.

-- MAA Reviews

I can only warmly recommend this book to students or as the basis for a course.

-- Monatshafte für Mathematik

The author presents in a very pleasant and readable way an introduction to ergodic theory for measure-preserving transformations of probability spaces. In my opinion, the book provides guidelines, classical examples and useful ideas for an introductory course in ergodic theory to students that have not necessarily already been taught Lebesgue measure theory.

-- Elemente der Mathematik

The book contains many (often easy or very easy) exercises, both in the text as well as at the end of each section.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Invitation to Ergodic Theory

- Cover Cover11 free
- Chapter 1. Introduction 112
- Chapter 2. Lebesgue measure 516
- Chapter 3. Recurrence and ergodicity 5970
- Chapter 4. The Lebesgue integral 131142
- Chapter 5. The ergodic theorem 175186
- Chapter 6. Mixing notions 201212
- Appendix A. Set notation and the completeness of ℝ 235246
- Appendix B. Topology of ℝ and metric spaces 241252
- Bibliographical notes 251262
- Bibliography 255266
- Back Cover Back Cover1274