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 $\delta_{x,n} = (\delta_x +\delta_{Tx}+\dots +\delta_{T^{n-1}x})/n$ 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 $\mu^*$ on the set of ergodic measures $\mathop{\mathrm{Erg}} X$ has the natural representation $\mu^*(A) =\mu(\{ x\in X_0 : \lim\delta_{x,n} \in A\})$, where $A\subset \mathop{\mathrm{Erg}} X$.