On almost specification and average shadowing properties

TL;DR

This paper explores the relationships between various shadowing and specification properties in dynamical systems, establishing implications, necessary conditions, and connections to other key dynamical notions.

Contribution

It clarifies the implications among almost specification, average shadowing, and related properties, and demonstrates the necessity of compactness for these implications.

Findings

Implications established between almost specification, average shadowing, and related properties.

Compactness is shown to be necessary for certain implications.

Limit shadowing in chain transitive systems implies shadowing.

Abstract

In this paper we study relations between almost specification property, asymptotic average shadowing property and average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples that compactness is a necessary condition for these implications to hold. As a consequence of our methodology we also obtain a proof that limit shadowing in chain transitive systems implies shadowing.