Recurrence, rigidity, and shadowing in dynamical systems
Volume 260 / 2023
Fundamenta Mathematicae 260 (2023), 263-279
MSC: Primary 37B20; Secondary 37B65, 37B45.
DOI: 10.4064/fm184-10-2022
Published online: 24 November 2022
Abstract
We examine the interplay between recurrence properties and the shadowing property in dynamical systems on compact metric spaces. In particular, 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.
