Mean dimension theory for infinite dimensional Bedford-McMullen sponges
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915049168175104 |
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| author | Huo, Qiang |
| author_facet | Huo, Qiang |
| contents | Tsukamoto (2022) introduced the notion of Bedford-McMullen carpet system, a subsystem of $([0,1]^{\mathbb{N}}\times[0,1]^{\mathbb{N}},shift)$ whose metric mean dimension and mean Hausdorff dimension does not coincide in general. The aim of this paper is to develop the mean dimension theory for Bedford-McMullen sponge system, which is a subsystem of $(([0,1]^r)^{\mathbb{N}},shift)$ with arbitrary $3\leq r\in\mathbb{N}$. In particular, we compute the metric mean dimension and mean Hausdorff dimension of such topological dynamical systems explicitly, extending the results by Tsukamoto. The metric mean dimension is a weighted combination of the standard topological entropy, whereas the mean Hausdorff dimension is expressed in terms of weighted topological entropy. We also exhibit a special situation for which the metric mean dimension and the mean Hausdorff dimension of a sponge system coincide. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_02278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mean dimension theory for infinite dimensional Bedford-McMullen sponges Huo, Qiang Dynamical Systems primary: 28A80, 37C45, 28D20, secondary: 37B40, 37A35 Tsukamoto (2022) introduced the notion of Bedford-McMullen carpet system, a subsystem of $([0,1]^{\mathbb{N}}\times[0,1]^{\mathbb{N}},shift)$ whose metric mean dimension and mean Hausdorff dimension does not coincide in general. The aim of this paper is to develop the mean dimension theory for Bedford-McMullen sponge system, which is a subsystem of $(([0,1]^r)^{\mathbb{N}},shift)$ with arbitrary $3\leq r\in\mathbb{N}$. In particular, we compute the metric mean dimension and mean Hausdorff dimension of such topological dynamical systems explicitly, extending the results by Tsukamoto. The metric mean dimension is a weighted combination of the standard topological entropy, whereas the mean Hausdorff dimension is expressed in terms of weighted topological entropy. We also exhibit a special situation for which the metric mean dimension and the mean Hausdorff dimension of a sponge system coincide. |
| title | Mean dimension theory for infinite dimensional Bedford-McMullen sponges |
| topic | Dynamical Systems primary: 28A80, 37C45, 28D20, secondary: 37B40, 37A35 |
| url | https://arxiv.org/abs/2412.02278 |