---
title: Ergodic Theory
description: Ergodic theory studies the statistical and long-term average behavior of dynamical systems using measure theory. Learners will understand how systems explore their state space over time, with applications to statistical mechanics and information theory.
category: mathematics
subcategory: dynamical-systems
difficulty: beginner, intermediate, advanced
url: /subject/ergodic-theory
---

# Ergodic Theory

Ergodic theory studies the statistical and long-term average behavior of dynamical systems using measure theory. Learners will understand how systems explore their state space over time, with applications to statistical mechanics and information theory.

## Available Resources

3 Books • 2 Websites

## Websites

### 1. Terence Tao's Ergodic Theory Notes

Terence Tao's blog posts and lecture notes on ergodic theory, including his graduate course on ergodic theory and its links to additive combinatorics. Readers work through recurrence theorems, Furstenberg correspondence, and measure-preserving systems with rigorous proofs.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://terrytao.wordpress.com/tag/ergodic-theory/

**Tags:** ergodic-theory, measure-preserving-systems, furstenberg-recurrence, additive-combinatorics, lecture-notes

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Books

### 1. An Introduction to Ergodic Theory

**Author:** Peter Walters

A graduate text from Springer's Graduate Texts in Mathematics series covering measure-preserving transformations, recurrence, ergodicity, mixing, the ergodic theorems, entropy, and topological dynamics. Readers gain the standard toolkit for studying invariant measures and entropy of dynamical systems.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0387951520?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, dynamical-systems

### 2. Ergodic Theory

**Author:** Karl E. Petersen

A Cambridge monograph presenting measure-theoretic ergodic theory, from the ergodic theorems and mixing properties to entropy, isomorphism theory, and Kakutani equivalence. Readers develop a rigorous understanding of how invariant measures classify the long-term behavior of transformations.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0521389976?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, dynamical-systems

### 3. Dynamical Systems and Ergodic Theory

**Author:** Mark Pollicott, Michiko Yuri

A London Mathematical Society student text introducing dynamical systems and ergodic theory through concrete examples such as circle rotations, expanding maps, and shifts. It covers recurrence, ergodic theorems, mixing, and entropy, preparing readers for applications in number theory.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0521575990?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, dynamical-systems

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