Introduction to Ergodic Theory

  title={Introduction to Ergodic Theory},
  author={Peter Walters},
Ergodic theory concerns with the study of the long-time behavior of a dynamical system. An interesting result known as Birkhoff’s ergodic theorem states that under certain conditions, the time average exists and is equal to the space average. The applications of ergodic theory are the main concern of this note. We will introduce fundamental concepts in ergodic theory, Birkhoff’s ergodic theorem and its consequences. 

Rudiments of Ergodic Theory

We give a short introduction to ergodic theory and its applications to topological dynamical systems. First we study the general properties of measure preserving transformations. Then we introduce

A Survey of Recent Results in the Spectral Theory of Ergodic Dynamical Systems

The purpose of this paper is to survey recent results in the spectral theory of ergodic dynamical systems. In addition we prove some known results using new methods and mention some new results,

Ergodic Theory: Interactions with Combinatorics and Number Theory

  • Thomas Ward
  • Mathematics
    Encyclopedia of Complexity and Systems Science
  • 2009
This article gives a brief overview of some of the ways in which number theory and combinatorics interacts with ergodic theory. The main themes are illustrated by examples related to recurrence,

Foundations of Ergodic Theory

Preface 1. Recurrence 2. Existence of invariant measures 3. Ergodic theorems 4. Ergodicity 5. Ergodic decomposition 6. Unique ergodicity 7. Correlations 8. Equivalent systems 9. Entropy 10.

A Note on Mean Ergodic Theory for Sums and Intersection of Lebesgue Spaces

Consideration is given to the mean ergodic theory for Lebesgue spaces. First, a necessary condition on Lebesgue spaces to satisfy mean ergodic theory is discussed. Then, the mean ergodic theory is

Ergodic theory of one dimensional Map

In this paper we study one dimensional linear and non-linear maps and its dynamical behavior. We study measure theoretical dynamical behavior of the maps. We study ergodic measure and Birkhoff

Realization of joint spectral radius via Ergodic theory

Based on the classic multiplicative ergodic theorem and the semi-uniform subadditive ergodic theorem, we show that there always exists at least one ergodic Borel probability measure such that the

Invariant measures for Markov maps of the interval

There is a theorem in ergodic theory which gives three conditions sufficient for a piecewise smooth mapping on the interval to admit a finite invariant ergodic measure equivalent to Lebesgue. When

Topological Wiener–Wintner ergodic theorems and a random L2 ergodic theorem

  • P. Walters
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1996
Abstract We give some topological ergodic theorems inspired by the Wiener-Wintner ergodic theorem. These theorems are used to give results for uniquely ergodic transformations and to study unique


Uniform distribution of sequences

( 1 ) {xn}z= Xn--Z_I Zin-Ztn-I is uniformly distributed mod 1, i.e., if ( 2 ) lim (1/N)A(x, N, {xn}z)-x (0x<_ 1), where A(x, N, {Xn)) denotes the number of indices n, l<=n<=N such that {x} is less