Shadowing, recurrence, and rigidity in dynamical systems

Abstract

In this paper we examine the interplay between recurrence properties and the shadowing property in dynamical systems on compact metric spaces. In particular, we demonstrate that if the dynamical system $(X,f)$ has shadowing, then it is recurrent if and only if it is minimal. Furthermore, we show that a uniformly rigid system $(X,f)$ has shadowing if and only if $X$ is totally disconnected and use this to demonstrate the existence of a space $X$ for which no surjective system $(X,f)$ has shadowing. We further refine these results to discuss the dynamics that can occur in spaces with compact space of self-maps.