So, let’s fix a compact metric space and a transformation which is continuous. We defined the space of probability Borel measures which are -invariant, showed it was nonempty, and proved that the extreme points correspond to ergodic measures (i.e. measures with respect to which is ergodic). We are interested in knowing what looks like, based solely on the **topological** properties of . Here are some techniques we can use:

1) If has no fixed points, then cannot have any atoms (i.e. ). Otherwise would have infinite measure.

2) The set of recurrent points in (i.e. such that there exists a sequence with ) has -measure one. We proved this earlier.

3) The set of non-wandering points has measure one. We define this notion now. Say that is **wandering** if there is a neighborhood of such that . In other words, the family of sets is disjoint. If not, say that is **non-wandering.** Any recurrent point, for instance, is non-wandering, which implies that the set of non-wandering points has measure one.

Here is an example. (more…)