# '22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

### "The mean orbital pseudo-metric in topological dynamics"

#### Pourmand, Habibeh

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.

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