An ergodic support of a dynamical system and a natural representation of Choquet distributions for invariant measures

Abstract

An ergodic support $X_0$ of a dynamical system $(X,T)$ with metrizable compact phase space $X$ is the set of all points $x\in X$ such that the corresponding sequence of empirical measures ${\delta }_{x,n}=({\delta }_x+{\delta }_{Tx}+\cdots +{\delta }_{T^{n-1}x})/n$ converges weakly to some ergodic measure. For every invariant probability measure $\mu$ on $X$ it is proven that $\mu (X_0)=1$ and Choquet distribution ${\mu }^{\ast }$ on the set of ergodic measures $\mathrm{Erg}X$ has the natural representation ${\mu }^{\ast }(A)=\mu (\{x\in X_0:\lim {\delta }_{x,n}\in A\})$, where $A\subset \mathrm{Erg}X$.

4 pages