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 $δ_{x,n} = (δ_x +δ_{Tx}+\dots +δ_{T^{n-1}x})/n$ converges weakly to some ergodic measure. For every invariant probability measure $μ$ on $X$ it is proven that $μ(X_0) =1$ and Choquet distribution $μ^*$ on the set of ergodic measures $\mathop{\mathrm{Erg}} X$ has the natural representation $μ^*(A) =μ(\{ x\in X_0 : \limδ_{x,n} \in A\})$, where $A\subset \mathop{\mathrm{Erg}} X$.
Fields of Study & Subjects
Stochastic processes and financial applications
Publication details
- Journal / source
- Mathematical Notes
- Publisher
- Pleiades Publishing Ltd
- Volume
- 120
- Issue
- 1-2
- Pages
- 392–395
- Publication date
- 1 August 2026
- Language
- EN