Topological dynamics for the endograph metric I: Equivalences with other metrics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917029205770240 |
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| author | López-Martínez, Antoni |
| author_facet | López-Martínez, Antoni |
| contents | Given a dynamical system $(X,f)$ we investigate several topological dynamical properties for its Zadeh extension $(\mathcal{F}(X),\hat{f})$ endowed with the endograph metric $d_{E}$. In particular, we prove that for topological $\mathcal{A}$-transitivity, topological $(\ell,\mathcal{A})$-recurrence, Devaney chaos, and for the specification property, the endograph metric behaves similarly to the supremum metric $d_{\infty}$, the Skorokhod metric $d_{0}$ and the sendograph metric $d_{S}$. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes in terms of point-$\mathcal{A}$-transitivity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_17990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Topological dynamics for the endograph metric I: Equivalences with other metrics López-Martínez, Antoni Dynamical Systems 37B02, 37B20, 47A16, 54A40, 54B20 Given a dynamical system $(X,f)$ we investigate several topological dynamical properties for its Zadeh extension $(\mathcal{F}(X),\hat{f})$ endowed with the endograph metric $d_{E}$. In particular, we prove that for topological $\mathcal{A}$-transitivity, topological $(\ell,\mathcal{A})$-recurrence, Devaney chaos, and for the specification property, the endograph metric behaves similarly to the supremum metric $d_{\infty}$, the Skorokhod metric $d_{0}$ and the sendograph metric $d_{S}$. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes in terms of point-$\mathcal{A}$-transitivity. |
| title | Topological dynamics for the endograph metric I: Equivalences with other metrics |
| topic | Dynamical Systems 37B02, 37B20, 47A16, 54A40, 54B20 |
| url | https://arxiv.org/abs/2510.17990 |