Various shadowing properties for time varying maps

TL;DR

This paper investigates various shadowing properties in time-varying dynamical systems, establishing their relationships, invariance under conjugacy, and conditions under which these properties hold, especially in expansive, contracting, and expanding cases.

Contribution

It introduces and analyzes multiple shadowing notions for time varying maps, exploring their interrelations and invariance, and characterizes conditions for their validity in different dynamical contexts.

Findings

Shadowing, h-shadowing, limit shadowing, s-limit shadowing, and exponential limit shadowing are conjugacy invariant.

Expansivity links shadowing properties, with shadowing implying other shadowing notions under this condition.

Contracting and expanding maps exhibit all studied shadowing properties.

Abstract

This paper is concerned with the study of various notions of shadowing of dynamical systems induced by a sequence of maps, so-called time varying maps, on a metric space. We define and study the shadowing, h-shadowing, limit shadowing, s-limit shadowing and exponential limit shadowing properties of these dynamical systems. We show that h-shadowing, limit shadowing and s-limit shadowing properties are conjugacy invariant. Also, we investigate the relationships between these notions of shadowing for time varying maps and examine the role that expansivity plays in shadowing properties of such dynamical systems. Specially, we prove some results linking s-limit shadowing property to limit shadowing property, and h-shadowing property to s-limit shadowing and limit shadowing properties. Moreover, under the assumption of expansivity, we show that the shadowing property implies the h-shadowing, s-limit shadowing and limit shadowing properties. Finally, it is proved that the uniformly contracting and uniformly expanding time varying maps exhibit the shadowing, limit shadowing, s-limit shadowing and exponential limit shadowing properties.