So, now it’s time to connect the topological notions of dynamical systems with ergodic theory (which makes use of measures). Our first example will use the notion of topological transitivity, which we now introduce. The next example will return to the story about recurrent points, which I talked a bit about yesterday.
Say that a homeomorphism of a compact metric space is topologically transitive if there exists with dense in . (For instance, a minimal homeomorphism is obviously topologically transitive.) Let be a countable basis for the topology of . Then the set of all such (with dense) is given by
In particular, if it is nonempty, then each is dense—being -invariant and containing —and this set is a dense by Baire’s theorem.
Proposition 1 Let have a Borel probability measure positive on every nonempty open set, and let be measure-preserving and ergodic. Then the set of with is of measure 1, so is topologically transitive.
Indeed, each must have measure zero or one by ergodicity, so measure 1 by hypothesis. Then take the countable intersection.
We now move to the abstract measure-theoretic framework, not topological.
Theorem 2 (Poincaré recurrence) Let be a measure-preserving transformation on a probability space . If is measurable, then there exists with such that for each , there is a sequence with .
In other words, points of are -frequently in . (more…)