Topology of global attractors for homeomorphisms with the topological shadowing property in $\mathbb{R}^m$

TL;DR

This paper investigates the conditions under which the shadowing property holds for dynamical systems with global attractors in non-compact spaces, revealing that it only occurs when the attractor is trivial.

Contribution

It establishes a precise characterization of when the shadowing property exists for systems with global attractors in non-compact spaces, linking it to the triviality of the attractor.

Findings

Shadowing property holds iff the global attractor is trivial.

Characterization of dynamical systems with shadowing in non-compact spaces.

Insight into the topology of attractors in non-compact dynamical systems.

Abstract

This paper studies the behavior of dynamical systems in non-compact spaces, specifically focusing on the concepts of global attractors and shadowing. Let $K$ be a compact global attractor. We show that the shadowing property holds in certain types of dynamical systems in non-compact spaces if and only if $K$ is trivial.