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arXiv · 2509.23669

A topological approach to fuzzy iterated function systems

Taras Banakh·Krzysztof Caban·Filip Strobin
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Abstract

In the paper we unify two extensions of the classical Hutchinson--Barnsley theory - the topological and the fuzzy-set approaches. We show that a fuzzy iterated function system (fuzzy IFS) on a Tychonoff space $X$ which is contracting w.r.t. some admissible multimetric, generates a natural fuzzy attractor in the hyperspace $\mathcal{K}_{\mathcal{F}}(X)$ of all compact fuzzy sets. As a consequence, we prove that a fuzzy IFS on a Hausdorff topological space which is topologically contracting admits a fuzzy attractor in a bit weaker sense. Our discussion involves investigations on topologies on the hyperspace $\mathcal{K}_{\mathcal{F}}(X)$ which are suitable for establishing convergence of sequences of iterations of a fuzzy Hutchinson operator.

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Taras Banakh, Krzysztof Caban, Filip Strobin. 2025-09-28. A topological approach to fuzzy iterated function systems. https://arxiv.org/abs/2509.23669

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