On the Space of Iterated Function Systems and Their Topological Stability

TL;DR

This paper explores the topological stability of iterated function systems (IFS) by analyzing their space structure, introducing a metric, and establishing conditions like shadowing properties that ensure stability.

Contribution

It characterizes the space of IFS as a hyperspace with the Hausdorff metric and links shadowing properties to topological stability in IFS.

Findings

Hausdorff distance is natural for IFS space

Shadowing property is necessary for stability

Shadowing plus expansiveness implies stability

Abstract

We study iterated function systems (IFS) with compact parameter space. We show that the space of IFS with phase space $X$ is the hyperspace of the space of self continuous maps of $X$. With this result we obtain that the Hausdorff distance is a natural metric for this space which we use to define topological stability. Then we prove, in the context of IFS, the classical results showing that shadowing property is a necessary condition for topological stability and shadowing property added to expansiveness are a sufficient condition for topological stability. To prove these statements, in fact, we use a stronger type of shadowing, called concordant shadowing property. We also give an example showing that concordant shadowing property is truly different than the traditional definition of shadowing property for IFS.