Unique ergodicity

Summary

A continuous map of a compact metric space is uniquely ergodic when it has exactly one invariant Borel probability measure. That measure is then automatically ergodic. For a uniquely ergodic map, time averages of every continuous function converge to the space average at every point, not only at almost every point. Unique ergodicity is the dynamical form of equidistribution, as in Weyl's equidistribution theorem.