A note on weak mean equicontinuity and strong mean sensitivity

Abstract: In this paper, we study the weak mean metric and give some properties by replacing the Besicovitch pseudometric with weak mean metric in the definition of mean equicontinuity and mean sensitivity. We study an opposite side of weak mean equicontinuity, strong mean sensitivity and we obtain a version of Auslander-Yorke dichotomies: minimal topological dynamical systems are either weak mean equicontinuous or strong mean sensitive, and transitive topological dynamical systemss are either almost weak mean equicontinuous or strong mean sensitive. Furthermore, motivated by the localized idea of sensitivity, we introduce some notions of new version sensitive tuples and study the properties of these sensitive tuples, we show that a transitive dynamical system is strong mean sensitive if and only if it admits a strong mean sensitive tuple. Finally, We introduce the notions of weakly mean equicontinuity of a topological dynamical system respect to a given continuous function $f$, and we show that a topological dynamical system is weakly mean equicontinuity then it is weakly mean equicontinuity with respect to every continuous function.