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

TL;DR

This paper characterizes the ergodic support of a dynamical system and introduces a natural representation of Choquet distributions for invariant measures, linking empirical measures to ergodic measures.

Contribution

It proves that the ergodic support has full measure for invariant measures and establishes a natural representation of Choquet distributions in terms of empirical measures.

Findings

Ergodic support has measure one for all invariant measures.

Choquet distribution can be represented via empirical measures converging to ergodic measures.

Provides a new perspective on the structure of invariant and ergodic 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 $A\subset \mathop{\mathrm{Erg}} X$.