Shadowing, asymptotic shadowing and s-limit shadowing

TL;DR

This paper investigates different shadowing concepts in dynamical systems, establishing equivalences for certain maps, constructing examples to show limitations, and exploring how these properties transfer to subsystems and extensions.

Contribution

It demonstrates the equivalence of classical and s-limit shadowing for specific maps, constructs examples separating shadowing notions, and analyzes transfer of shadowing properties.

Findings

Classical and s-limit shadowing coincide for tent maps.

Constructed systems with shadowing but not limit shadowing.

Shadowing properties transfer to subsystems and inverse limits.

Abstract

We study three notions of shadowing: classical shadowing, limit (or asymptotic) shadowing, and s-limit shadowing. We show that classical and s-limit shadowing coincide for tent maps and, more generally, for piecewise linear interval maps with constant slopes, and are further equivalent to the linking property introduced by Chen in 1991. We also construct a system which exhibits shadowing but not limit shadowing, and we study how shadowing properties transfer to maximal transitive subsystems and inverse limits (sometimes called natural extensions). Where practicable, we show that our results are best possible by means of examples.