On the shadowing and limit shadowing properties

TL;DR

This paper investigates the relationship between shadowing and limit shadowing properties in dynamical systems, establishing key implications and equivalences, especially for equicontinuous maps on compact metric spaces.

Contribution

It proves that limit shadowing implies shadowing on the non-wandering set and shows the equivalence of the two properties for equicontinuous maps.

Findings

Limit shadowing implies shadowing on the non-wandering set.

Shadowing and limit shadowing are equivalent for equicontinuous maps.

The results deepen understanding of shadowing properties in dynamical systems.

Abstract

We study the relation between the shadowing property and the limit shadowing property. We prove that if a continuous self-map $f$ of a compact metric space has the limit shadowing property, then the restriction of $f$ to the non-wandering set satisfies the shadowing property. As an application, we prove the equivalence of the two shadowing properties for equicontinuous maps.