A First Course in Ergodic Theory

Karma Dajani, Charlene Kalle

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN:
  9781032021843
  9780367226206
• Edition: 1
• eBook ISBN:
  9781000402780
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

A First Course in Ergodic Theory provides readers with an introductory course in Ergodic Theory. This textbook has been developed from the authors' own notes on the subject, which they have been teaching since the 1990s. Over the years they have added topics, theorems, examples and explanations from various sources. The result is a book that is easy to teach from and easy to learn from — designed to require only minimal prerequisites.

Features

• Suitable for readers with only a basic knowledge of measure theory, some topology and a very basic knowledge of functional analysis
• Perfect as the primary textbook for a course in Ergodic Theory
• Examples are described and are studied in detail when new properties are presented.

CHAPTER 1

Measure Preservingness and Basic Examples

1.1 WHAT IS ERGODIC THEORY?

Dynamical systems theory is the area of mathematics dedicated to the study of processes that evolve over time. There are many ways to approach this subject and ergodic theory focuses on describing the asymptotic average behavior of dynamical systems. To do that it uses techniques from many different fields including probability theory, statistical mechanics, number theory, vector fields on manifolds, group actions of homogeneous spaces and fractal geometry.

The word ergodic is a mixture of two Greek words: ergon (work) and odos (path). The word was introduced by L. Boltzmann (1844–1906) regarding his hypothesis in statistical mechanics: for large systems of interacting particles in equilibrium, the time average along a single trajectory equals the space average. The hypothesis as he stated it was false, and the search for conditions under which these two quantities are equal led to the birth of ergodic theory as it is known nowadays.

A dynamical system consists of a space X that represents the collection of all states the system can be in, called state space, and a way to describe th...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Author Bios
• CHAPTER 1 ▪ Measure Preservingness and Basic Examples
• CHAPTER 2 Recurrence and Ergodicity
• CHAPTER 3 ▪ The Pointwise Ergodic Theorem and Mixing
• CHAPTER 4 ▪ More Ergodic Theorems
• CHAPTER 5 ▪ Isomorphisms and Factor Maps
• CHAPTER 6 ▪ The Perron-Frobenius Operator
• CHAPTER 7 ▪ Invariant Measures for Continuous Transformations
• CHAPTER 8 ▪ Continued Fractions
• CHAPTER 9 ▪ Entropy
• CHAPTER 10 ▪ The Variational Principle
• CHAPTER 11 ▪ Infinite Ergodic Theory
• CHAPTER 12 ▪ Appendix
• Bibliography
• Index