Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems
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arXiv
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| Format: | Preprint |
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2024
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| author | Alvarez, Illych López-Martínez, Antoni Peris, Alfred |
| author_facet | Alvarez, Illych López-Martínez, Antoni Peris, Alfred |
| contents | We study for a dynamical system $f:X\longrightarrow X$ some of the principal topological recurrence-kind properties with respect to the induced maps $\overline{f}:\mathcal{K}(X)\longrightarrow\mathcal{K}(X)$, on the hyperspace of non-empty compact subsets of $X$, and $\hat{f}:\mathcal{F}(X)\longrightarrow\mathcal{F}(X)$, on the space of normal fuzzy sets consisting of the upper-semicontinuous functions $u:X\longrightarrow [0,1]$ with compact support and such that $u^{-1}(\{1\})\neq\varnothing$. In particular, we characterize the properties of topological and multiple recurrence for the extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$, which cover the cases of the so-called nonwandering and Van der Waerden systems. Special attention is given to the case where the underlying space is completely metrizable, for which we obtain some stronger point-recurrence equivalences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15826 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems Alvarez, Illych López-Martínez, Antoni Peris, Alfred Dynamical Systems 37B02, 37B20, 54A40, 54B20 We study for a dynamical system $f:X\longrightarrow X$ some of the principal topological recurrence-kind properties with respect to the induced maps $\overline{f}:\mathcal{K}(X)\longrightarrow\mathcal{K}(X)$, on the hyperspace of non-empty compact subsets of $X$, and $\hat{f}:\mathcal{F}(X)\longrightarrow\mathcal{F}(X)$, on the space of normal fuzzy sets consisting of the upper-semicontinuous functions $u:X\longrightarrow [0,1]$ with compact support and such that $u^{-1}(\{1\})\neq\varnothing$. In particular, we characterize the properties of topological and multiple recurrence for the extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$, which cover the cases of the so-called nonwandering and Van der Waerden systems. Special attention is given to the case where the underlying space is completely metrizable, for which we obtain some stronger point-recurrence equivalences. |
| title | Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems |
| topic | Dynamical Systems 37B02, 37B20, 54A40, 54B20 |
| url | https://arxiv.org/abs/2406.15826 |