  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.
