We study properties and applications of the mean orbital pseudo-metric $\bar{\rho}$ on a topological dynamical system $(X,T)$ defined by \[ \bar{\rho}(x,y)= \limsup_{n\to \infty} \min_{\sigma \in S_n} \frac{1}{n}\sum_{k=0}^{n-1} d(T^k(x), T^{\sigma(k)}(y)), \] where $x,y\in X$, $d$ is a metric for $X$, and $S_n$ is the permutation group of the set $\{0,1,\ldots,n-1\}$. Writing $\hat{\omega}(x)$ for the set of $T$-invariant measure generated by the orbit of a point $x\in X$, we prove that the function $x\mapsto \hat{\omega}(x)$ is $\bar{\rho}$ uniformly continuous. This allows us to characterise equicontinuity with respect to the mean orbital pseudo-metric ($\bar{\rho}$-equicontinuity) and connect it to such notions as uniform or continuously pointwise ergodic systems studied recently by Downarowicz and Weiss. This is joint work with F. Cai, D. Kwietniak, and J. Li.

[IPM Youth Seminars on Topology and Dynamics]
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