A new metric for statistical properties of long time behaviors

Abstract

Let $(X,T)$ be a topological dynamical system with metric $d$. We define a new function $\overline{F}(x,y)=\underset{n\to +\infty }{\mathrm{limsup}}\underset{\sigma \in S_n}{\inf }\frac{1}{n}\sum _{k=1}^{n}d(T^{k}x,T^{\sigma (k)}y)$ by using permutation group $S_n$. It's shown $F(x,y)=\underset{n\to +\infty }{\lim }\underset{\sigma \in S_n}{\inf }\frac{1}{n}\sum _{k=1}^{n}d(T^{k}x,T^{\sigma (k)}y)$ 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 $f^{\ast }(x)=\underset{n\to +\infty }{\lim }\frac{1}{n}\sum _{k=1}^{n}f(T^{k}x)$ exist and are continuous for all $f\in C(X)$.