Li-Yorke chaos on fuzzy dynamical systems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918166912827392 |
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| author | Álvarez, Illych López-Martínez, Antoni |
| author_facet | Álvarez, Illych López-Martínez, Antoni |
| contents | Given a dynamical system $(X,f)$ we investigate how several variants of Li-Yorke chaos behave with respect to the extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$, where $\overline{f}$ is the hyperextension of $f$ acting on the space $\mathcal{K}(X)$ of non-empty compact subsets of $X$, and where $\hat{f}$ denotes the Zadeh extension of $f$ acting on the space $\mathcal{F}(X)$ of normal fuzzy subsets of $X$. We first prove that the main variants of Li-Yorke chaos transfer from $(X,f)$ to $(\mathcal{K}(X),\overline{f})$ and from $(\mathcal{K}(X),\overline{f})$ to $(\mathcal{F}(X),\hat{f})$, but that the converse implications do not hold in general. However, combining the notions of proximality and sensitivity we introduce Cantor-dense Li-Yorke chaos, and we prove that this strengthened variant of chaos does transfer from $(\mathcal{F}(X),\hat{f})$ to $(\mathcal{K}(X),\overline{f})$ under natural assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20240 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Li-Yorke chaos on fuzzy dynamical systems Álvarez, Illych López-Martínez, Antoni Dynamical Systems 37B02, 37B05, 37B25, 54A40, 54B20 Given a dynamical system $(X,f)$ we investigate how several variants of Li-Yorke chaos behave with respect to the extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$, where $\overline{f}$ is the hyperextension of $f$ acting on the space $\mathcal{K}(X)$ of non-empty compact subsets of $X$, and where $\hat{f}$ denotes the Zadeh extension of $f$ acting on the space $\mathcal{F}(X)$ of normal fuzzy subsets of $X$. We first prove that the main variants of Li-Yorke chaos transfer from $(X,f)$ to $(\mathcal{K}(X),\overline{f})$ and from $(\mathcal{K}(X),\overline{f})$ to $(\mathcal{F}(X),\hat{f})$, but that the converse implications do not hold in general. However, combining the notions of proximality and sensitivity we introduce Cantor-dense Li-Yorke chaos, and we prove that this strengthened variant of chaos does transfer from $(\mathcal{F}(X),\hat{f})$ to $(\mathcal{K}(X),\overline{f})$ under natural assumptions. |
| title | Li-Yorke chaos on fuzzy dynamical systems |
| topic | Dynamical Systems 37B02, 37B05, 37B25, 54A40, 54B20 |
| url | https://arxiv.org/abs/2510.20240 |