A note on weak Banach mean equicoontinuity

Consider a topological dynamical system $(X,T)$ endowed with the metric $d$. We introduce a novel function as $\overline{BF}(x,y)={\mathrm{lim\hspace{0.17em}sup}}_{n-m\to +\mathrm{\infty }}{\inf }_{\sigma \in S_{n,m}}\frac{1}{n-m}{\sum }_{k=m}^{n-1}d(T^{k}x,T^{\sigma (k)}y)$, where the permutation group $S_{n,m}$ is utilized. It is demonstrated that $BF(x,y)$ exists when $x,y\in X$ are uniformly generic points. Leveraging this function, we introduce the concept of weak Banach mean equicontinuity and establish that the dynamical system $(X,T)$ exhibits weak Banach mean equicontinuity if and only if the uniform time averages $f_B^{\ast }(x)={\lim }_{n-m\to +\mathrm{\infty }}\frac{1}{n-m}{\sum }_{k=m}^{n-1}f\left(T^{k}x\right)$ are continuous for all $f\in C(X)$. Finally, we demonstrate that in the case of a transitive system, the equivalence between weak Banach mean equicontinuity and weak mean equicontinuity is established.