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 .
To see this, consider the sets
Then and since . Set and . Then
and this latter set is the limit of the increasing sequence of sets , each of which has measure zero. This proves the theorem.
Topological application of Poincaré recurrence
Let be a compact metric space, a continuous map. Say that is -recurrent if there exists a sequence with . Birkhoff’s theorem states that there always exists a -recurrent point if is a homeomorphism.
Theorem 3 If preserves the probability measure , then almost all points of are -recurrent.
Let be a countable basis. Then is not -recurrent if for some and , as is easily seen (check the two cases a periodic point and non-periodic separately). So
However, each set in this union has -measure zero by Poincaré’s theorem, which proves this one.
We will show that any continuous has a probability measure with respect to which is measure-preserving, so
Corollary 4 If is a continuous transformation of the compact metric space , then -recurrent points exist.