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 $\overline{F}(x,y)=\limsup\limits_{n \to +\infty} \inf\limits_{\sigma \in S_n} \frac 1n \sum\limits_{k=1}^n d(T^k x,T^{\sigma(k)} y)$byusingpermutationgroup$S_n$\mathrm{.}I{t}^{\mathrm{\prime }}sshown$F(x,y)=\lim\limits_{n \to +\infty} \inf\limits_{\sigma \in S_n} \frac 1n \sum\limits_{k=1}^n d(T^k x,T^{\sigma(k)} y)$existswhen$x,y \in X$ are generic points. Applying this function, we prove $(X,T)$ is uniquely ergodic if and only if $\bar{F}(x,y)=0$ for any $x,y\in X$. The characterizations of ergodic measures and physical measures by $\bar{F}(x,y)$ 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 $f^{*}(x)=\lim\limits_{n \to +\infty}\frac 1n \sum\limits_{k=1}^n f(T^k x)$existandarecontinuousforall$f \in C(X)$.