A topological approach to fuzzy iterated function systems
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909812998012928 |
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| author | Banakh, Taras Caban, Krzysztof Strobin, Filip |
| author_facet | Banakh, Taras Caban, Krzysztof Strobin, Filip |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_23669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A topological approach to fuzzy iterated function systems Banakh, Taras Caban, Krzysztof Strobin, Filip Dynamical Systems 28A80, 47H09, 03E72 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. |
| title | A topological approach to fuzzy iterated function systems |
| topic | Dynamical Systems 28A80, 47H09, 03E72 |
| url | https://arxiv.org/abs/2509.23669 |