Ergodic transformation

Summary

A measure-preserving transformation of a probability space is ergodic if it has no invariant measurable set other than null sets and sets of full measure. Ergodic maps cannot be split into two separate systems of positive measure. By Birkhoff's ergodic theorem, for an ergodic map the time average of an integrable function along almost every orbit equals its space average. This is the mathematical form of the ergodic hypothesis of statistical mechanics.