Consider a topological dynamical system $(X, T)$ endowed with the metric $d$. We introduce a novel function as $\overline{ BF}(x, y) = \limsup_{ n-m \rightarrow +\infty} \inf_{ \sigma \in S_{ n,m}} \frac{ 1}{ n-m} \sum_{ k=m}^{ n-1} d\left(T^{ k} x, T^{ \sigma(k)} y\right)$, 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 $f_B^{ *}(x) = \lim_{ n-m \rightarrow +\infty} \frac{ 1}{ n-m} \sum_{ k=m}^{ n-1} f\left(T^{ k} x\right)$ 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.
Categories: math.DS