A note on weak Banach mean equicoontinuity

2024/01/18 by Yang, Zhongxuan, Huang, Xiaojun
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.09744

Abstract

Consider a topological dynamical system (X, T) endowed with the metric d. We introduce a novel function as BF(x, y) = \limsup_{n-m → +∞} inf_σ∈ S_n,m (1)/(n-m) ∑_k=m^n-1 d(T^k x, T^σ(k) y), where the permutation group S_n,m is utilized. It is demonstrated that BF(x, y) exists when x, y ∈ 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 → +∞} (1)/(n-m) ∑_k=m^n-1 f(T^k x) are continuous for all f ∈ 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.

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