Ergodic

In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. An older term was 'metrically transitive. Ergodic theory, the study of ergodic transformations, grew out of an attempt to prove the ergodic hypothesis of statistical physics.

Consider the "time average" of a well-behaved function f. This is defined as the average (if it exists) over iterations of T starting from some initial point x.

Consider also the "space average" or "phase average" of f, defined as

where μ is the measure of the probability space.

In general the time mean and space mean may be different. For an ergodic transformation, the time mean is equal to the space mean almost everywhere. This is the celebrated ergodic theorem, in an abstract form due to G. D. Birkhoff.