Shadowing property for set-valued map and its inverse limit

TL;DR

This paper explores the connection between the shadowing property of set-valued maps and their inverse limit systems, establishing conditions under which shadowing is preserved and equivalent.

Contribution

It proves the equivalence of shadowing properties between set-valued maps and their inverse limit systems under certain conditions.

Findings

Shadowing property of set-valued maps is equivalent to that of their inverse limit systems.

Expansiveness and openness of set-valued maps imply shadowing in inverse limit systems.

The paper establishes necessary and sufficient conditions for shadowing in set-valued dynamics.

Abstract

In this article, we investigate the relationship between the shadowing property of set-valued maps and their associated inverse limit systems. We show that if a set-valued map is expansive and open in the context of set-valued dynamics, then certain induced inverse limit systems have the shadowing property. Additionally, we prove that a continuous set-valued map has the shadowing property if and only if some of its induced inverse limit system also has shadowing property. Finally, we establish that the shadowing property of a set-valued map is equivalent to the shadowing property of its induced inverse set-valued system.