A note on shadowing with chain transitivity

Abstract

Let f : X→X be a continuous map of a compact metric space X. The map f induces in a natural way a map fM on the space M(X) of probability measures on X, and a transformation fK on the space K(X) of closed subsets of X. In this paper, we show that if (X,f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:(a)fn and (fK)n are syndetically sensitive for all n⩾1.(b)fn and (fK)n are equicontinuous for all n⩾1.In particular, we show that for a continuous map f : X→X of a compact metric space X with infinite elements, if f is a chain transitive map with the shadowing property, then fn and (fK)n are syndetically sensitive for all n⩾1. Also, we show that if fM (resp. fK) is chain transitive and syndetically sensitive, and fM (resp. fK) has the shadowing property, then f is sensitive.In addition, we introduce the notion of ergodical sensitivity and present a sufficient condition for a chain transitive system (X,f) (resp. (M(X),fM)) to be ergodically sensitive. As an application, we show that for a L-hyperbolic homeomorphism f of a compact metric space X, if f has the AASP, then fn is syndetically sensitive and multi-sensitive for all n⩾1.