On the properties of the mean orbital pseudo-metric

Abstract

Given a topological dynamical system (X,T), we study properties of the mean orbital pseudo-metric E¯ defined byE¯(x,y)=limsupn→∞minσ∈Sn⁡1n∑k=0n−1d(Tk(x),Tσ(k)(y)), where x,y∈X and Sn is the permutation group of {0,1,…,n−1}. Let ωˆT(x) denote the set of measures quasi-generated by a point x∈X. We show that the map x↦ωˆT(x) is uniformly continuous if X is endowed with the pseudo-metric E¯ and the space of compact subsets of the set of invariant measures is considered with the Hausdorff distance. We also obtain a new characterisation of E¯-continuity, which connects it to other properties studied in the literature, like continuous pointwise ergodicity introduced by Downarowicz and Weiss. Finally, we apply our results to reprove some known results on E¯-continuous and mean equicontinuous systems.