Generalized entropy of induced zero-entropy systems
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917971590381568 |
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| author | Lacerda, Gabriel |
| author_facet | Lacerda, Gabriel |
| contents | Given a compact metric space $X$ and a continuous map $T: X \to X$, the induced hyperspace map $T_\mathcal{K}$ acts on the hyperspace $\mathcal{K}(X)$ of nonempty closed sets of $X$, and the measure-induced map $T_*$ acts on the space of probability measures $\mathcal{M}(X)$. It is proven that a large class of zero-entropy dynamical systems exhibits infinite metric mean dimension in its induced hyperspace map $T_\mathcal{K}$. This work also builds on the concept of generalized entropy, which is fundamental for studying the complexity of zero-entropy systems. Lower bounds of the generalized entropy of the measure-induced map $T_*$ are established, assuming that the base system $T$ has zero topological entropy. Moreover, upper bounds of the generalized entropy are explicitly computed for the measure-induced map of the Morse-Smale diffeomorphisms on the circle. Finally, it is shown that the generalized entropy of $T_*$ is a lower bound for the generalized entropy of $T_\mathcal{K}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22944 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized entropy of induced zero-entropy systems Lacerda, Gabriel Dynamical Systems 37B02, 54B20 Given a compact metric space $X$ and a continuous map $T: X \to X$, the induced hyperspace map $T_\mathcal{K}$ acts on the hyperspace $\mathcal{K}(X)$ of nonempty closed sets of $X$, and the measure-induced map $T_*$ acts on the space of probability measures $\mathcal{M}(X)$. It is proven that a large class of zero-entropy dynamical systems exhibits infinite metric mean dimension in its induced hyperspace map $T_\mathcal{K}$. This work also builds on the concept of generalized entropy, which is fundamental for studying the complexity of zero-entropy systems. Lower bounds of the generalized entropy of the measure-induced map $T_*$ are established, assuming that the base system $T$ has zero topological entropy. Moreover, upper bounds of the generalized entropy are explicitly computed for the measure-induced map of the Morse-Smale diffeomorphisms on the circle. Finally, it is shown that the generalized entropy of $T_*$ is a lower bound for the generalized entropy of $T_\mathcal{K}$. |
| title | Generalized entropy of induced zero-entropy systems |
| topic | Dynamical Systems 37B02, 54B20 |
| url | https://arxiv.org/abs/2503.22944 |