A topological dynamical system on a compact metric space is said to be uniquely ergodic if there is exactly one invariant Borel probability measure (and hence also exactly one ergodic Borel probability measure).

An irrational circle rotation is uniquely ergodic, where the unique invariant measure is the Lebesgue measure.

In this lecture we show that a stronger version of the Birkhoff Ergodic Theorem holds for uniquely ergodic systems. (Spoiler: for uniquely ergodic systems, time average = space average for all points, rather than just for almost all points.)



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