Liqi Zheng | Searcharxiv

arXiv2021

Weak mean equicontinuity for a countable discrete amenable group action

The weak mean equicontinuous properties for a countable discrete amenable group $G$ acting continuously on a compact metrizable space $X$ are studied. It is shown that the weak mean equicontinuity of $(X \times X,G)$ is equivalent to the mean equicontinuity of $(X,G)$. Moreover, when $(X,G)$ has full measure center or $G$ is abelian, it is shown that $(X,G)$ is weak mean equicontinuous if and only if all points in $X$ are uniquely ergodic points and the map $x \to μ_x^G$ is continuous, where $μ_x^G$ is the unique ergodic measure on $\{\ol{Orb(x)}, G\}$.

math.DS↗

arXiv2020

A new metric for statistical properties of long time behaviors

Let $(X,T)$ be a topological dynamical system with metric $d$. We define a new function $\overline{F}(x,y)=\limsup\limits_{n \to +\infty} \inf\limits_{σ\in S_n} \frac 1n \sum\limits_{k=1}^n d(T^k x,T^{σ(k)} y)$ by using permutation group $S_n$. It's shown $F(x,y)=\lim\limits_{n \to +\infty} \inf\limits_{σ\in S_n} \frac 1n \sum\limits_{k=1}^n d(T^k x,T^{σ(k)} y)$ exists when $x,y \in X$ are generic points. Applying this function, we prove $(X,T)$ is uniquely ergodic if and only if $\overline{F}(x,y)=0$ for any $x,y \in X$. The characterizations of ergodic measures and physical measures by $\overline{F}(x,y)$ are given. We introduce the notion of weak mean equicontinuity and prove that $(X,T)$ is weak mean equicontinuous if and only if the time averages $f^{*}(x)=\lim\limits_{n \to +\infty}\frac 1n \sum\limits_{k=1}^n f(T^k x)$ exist and are continuous for all $f \in C(X)$.

math.DS↗