Mean dimension explosion of induced homeomorphisms
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908589989298176 |
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| author | Lacerda, Gabriel Romaña, Sergio |
| author_facet | Lacerda, Gabriel Romaña, Sergio |
| contents | Given $X$ a compact metric space and $T: X \to X$ a continuous map, the induced hyperspace map $T_\mathcal{K}$ acts on the hyperspace $\mathcal{K}(X)$ of closed and nonempty subsets of $X$, and on the continuum hyperspace $\mathcal{C}(X) \subset \mathcal{K}(X)$ of connected sets. This work studies the mean dimension explosion phenomenon: when the base system $T$ has zero topological entropy, but the mean dimension of the induced map $T_\mathcal{K}$ is infinite. In particular, this phenomenon occurs for Morse-Smale diffeomorphisms. Furthermore, for a circle homeomorphism $H$, the mean dimension explosion does not occur if and only if $H$ is conjugate to a rotation. For the metric mean dimension, a different result is obtained: we establish sufficient conditions for the induced hyperspace map to have zero or infinite metric mean dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08146 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mean dimension explosion of induced homeomorphisms Lacerda, Gabriel Romaña, Sergio Dynamical Systems General Topology Given $X$ a compact metric space and $T: X \to X$ a continuous map, the induced hyperspace map $T_\mathcal{K}$ acts on the hyperspace $\mathcal{K}(X)$ of closed and nonempty subsets of $X$, and on the continuum hyperspace $\mathcal{C}(X) \subset \mathcal{K}(X)$ of connected sets. This work studies the mean dimension explosion phenomenon: when the base system $T$ has zero topological entropy, but the mean dimension of the induced map $T_\mathcal{K}$ is infinite. In particular, this phenomenon occurs for Morse-Smale diffeomorphisms. Furthermore, for a circle homeomorphism $H$, the mean dimension explosion does not occur if and only if $H$ is conjugate to a rotation. For the metric mean dimension, a different result is obtained: we establish sufficient conditions for the induced hyperspace map to have zero or infinite metric mean dimension. |
| title | Mean dimension explosion of induced homeomorphisms |
| topic | Dynamical Systems General Topology |
| url | https://arxiv.org/abs/2404.08146 |