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

2023/09/19 by Bakhtin, V. I.
#37A05 #46A55 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.10573

Abstract

An ergodic support X₀ of a dynamical system (X,T) with metrizable compact phase space X is the set of all points x∈ X such that the corresponding sequence of empirical measures δ_x,n = (δ_x +δ_Tx+… +δ_T^n-1x)/n converges weakly to some ergodic measure. For every invariant probability measure μ on X it is proven that μ(X₀) =1 and Choquet distribution μ^* on the set of ergodic measures \mathopErg X has the natural representation μ^*(A) =μ(\ x∈ X₀ : limδ_x,n ∈ A\), where A⊂ \mathopErg X.

Related

• On density of ergodic measures and generic points
• Even the vague specification property implies density of ergodic measures
• Choquet simplices as spaces of invariant probability measures of post-critical sets
• When all points are generic for ergodic measures
• On the closedness of ergodic measures in a characteristic class