Shadowing, recurrence, and rigidity in dynamical systems

TL;DR

This paper explores the relationship between shadowing, recurrence, and rigidity in dynamical systems, revealing conditions under which these properties coincide or imply specific space structures.

Contribution

It establishes new equivalences between shadowing and recurrence/minimality, and characterizes systems with shadowing in rigid and totally disconnected spaces.

Findings

Shadowing systems are recurrent iff they are minimal.

Rigid systems have shadowing iff the space is totally disconnected.

There exist spaces where no surjective system has shadowing.

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.