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arXiv · 2009.14317

On the Space of Iterated Function Systems and Their Topological Stability

Alexander Arbieto·Alexandre Trilles
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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.

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Alexander Arbieto, Alexandre Trilles. 2020-09-29. On the Space of Iterated Function Systems and Their Topological Stability. https://arxiv.org/abs/2009.14317

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