Shadowing relations with structural and topological stability in iterated function systems

TL;DR

This paper explores the relationship between shadowing, stability, and expansiveness in iterated function systems, establishing conditions under which these properties imply each other and providing counterexamples.

Contribution

It formulates definitions for stability and expansiveness in IFSs and proves that shadowing is necessary for stability, with conditions for their equivalence.

Findings

Shadowing property is necessary for structural stability in IFSs.

Structural stability implies shadowing property in IFSs.

Counterexample shows the converse does not always hold.

Abstract

This paper aims at formulating definitions of topological stability, structural stability, and expansiveness property for an iterated function system( abbrev, IFS). It is going to show that the shadowing property is necessary condition for structural stability in IFSs. Then, it proves the previous converse demonstration with the addition of expansiveness property for IFSs. It asserts that structural stability implies shadowing property in IFSs and presents an example to reject of the converse assertion.