On the properties of the mean orbital pseudo-metric

Given a topological dynamical system $(X,T)$, we study properties of the mean orbital pseudo-metric $\bar{E}$ defined by $$\bar{E}(x,y)=\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\underset{\sigma \in S_n}{\min }\frac{1}{n}\sum _{k=0}^{n-1}d(T^k(x),T^{\sigma (k)}(y)),$$ where $x,y\in X$ and $S_n$ is the permutation group of $\{0,1,\dots ,n-1\}$. Let ${\widehat{\omega }}_T(x)$ denote the set of measures quasi-generated by a point $x\in X$. We show that the map $x\mapsto {\widehat{\omega }}_T(x)$ is uniformly continuous if $X$ is endowed with the pseudo-metric $\bar{E}$ 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 $\bar{E}$-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 $\bar{E}$-continuous and mean equicontinuous systems.