On the Space of Iterated Function Systems and Their Topological Stability

We study iterated function systems with compact parameter space (IFS for short). We show that the space of IFS with phase space X is the hyperspace of the space of continuous maps from X to itself, which allows us to use the Hausdorff metric to define topological stability for IFS. We then prove that the concordant shadowing property is a necessary condition for topological stability and it is a sufficient condition if the IFS is expansive. Additionally, we provide an example to show that the concordant shadowing property is genuinely different from the traditional notion that, in our setting, becomes too weak.