The research of mean equicontinuity, mean sensitivity, and shadowing property

Abstract

We study the dynamical properties of mean equicontinuity, mean sensitivity, tracking property, and periodic points set in the hyperspace of uniform space. Let (Y,ξ) be uniform space, T:Y→Y be uniformly continuous and (C(Y),Cξ) be hyperspace of (Y,ξ).Then, we can get some conclusions: (a) T is mean equicontinous if and only if the induced map CT is mean equicontinous; (b) if the induced map CT is mean sensitive, then T is mean sensitive; (c) the induced map CT has tracking property implying that T has tracking property; (d) P(T) is dense in Y implying that P(CT) is dense in C(Y). In addition, we also study dynamical property of (G,h)− tracking property in metric G− space, and prove that if the map T has (G,h)− tracking property, then, for any k>1, the map Tk has (G,h)− tracking property.