Unique ergodicity - Knowlpedia

Definition

A continuous map $T:X\to X$ of a compact space is uniquely ergodic if it has exactly one $T$ -invariant Borel probability measure. A continuous flow $(\varphi_t)_{t\in\mathbb R}$ is uniquely ergodic if there is exactly one probability measure invariant under every $\varphi_t$ .

Uniform time averages

If $T$ is uniquely ergodic with invariant measure $\mu$ , then for every continuous $f:X\to\mathbb R$ ,

\frac1N\sum_{k=0}^{N-1}f(T^kx) \longrightarrow \int_Xf\,d\mu

uniformly in $x$ . An analogous statement holds for continuous-time averages of a uniquely ergodic flow.

Warning

Unique ergodicity is stronger than ergodicity with respect to one chosen measure. It is a topological assertion about the uniqueness of all invariant probability measures.

References

Peter Walters, An Introduction to Ergodic Theory, Springer, 1982. DOI record.