So, let’s fix a compact metric space {X} and a transformation {T: X \rightarrow X} which is continuous. We defined the space {M(X,T)} of probability Borel measures which are {T}-invariant, showed it was nonempty, and proved that the extreme points correspond to ergodic measures (i.e. measures with respect to which {T} is ergodic). We are interested in knowing what {M(X,T)} looks like, based solely on the topological properties of {T}. Here are some techniques we can use:

1) If {T} has no fixed points, then {\mu \in M(X,T)} cannot have any atoms (i.e. {\mu(\{x\})=0, x \in X}). Otherwise {\{x, Tx , T^2x, \dots \}} would have infinite measure.

2) The set of recurrent points in {X} (i.e. {x \in X} such that there exists a sequence {n_i \rightarrow \infty} with {T^{n_i}x \rightarrow x}) has {\mu}-measure one. We proved this earlier.

3) The set of non-wandering points has measure one. We define this notion now. Say that {x \in X} is wandering if there is a neighborhood {U} of {X} such that {T^{-n}(U) \cap U = \emptyset, \forall n \in \mathbb{N}}. In other words, the family of sets {T^{i}(U), i \in \mathbb{Z}_{\geq 0}} is disjoint. If not, say that {x} 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…)