Ergodicity

Abstract

Around the turn of the century the work of Boltzmann and Gibbs on statistical mechanics raised a mathematical problem which, in our context, can be stated as follows: given a measure-preserving map of a space (X, A, μ) and an integrable function f: X → R, find conditions under which the limit

$$\mathop {\lim }\limits_{n \to + \infty } \frac{{f\left( x \right) + f\left( {T\left( x \right)} \right) + \cdots + f\left( {{T^{n - 1}}\left( x \right)} \right)}}{n}$$

(1)

exists and is constant almost everywhere. Similar questions had already shown up in other areas of mathematics, for example, in the problem of the average movement of the perihelion in celestial mechanics (see Arnold [A6]).