Mean dimension and rate-distortion function revisited
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912638615683072 |
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| author | Yang, Rui |
| author_facet | Yang, Rui |
| contents | Around the mean dimensions and rate-distortion functions, using some tools from local entropy theory this paper establishes the following main results:
$(1)$ We prove that for non-ergodic measures associated with almost sure processes, the mean Rényi information dimension coincides with the information dimension rate. This answers a question posed by Gutman and Śpiewak (in Around the variational principle for metric mean dimension, \emph{Studia Math.} \textbf{261}(2021) 345-360).
$(2)$ We introduce four types of rate-distortion entropies and establish their relation with Kolmogorov-Sinai entropy.
$(3)$ We show that for systems with the marker property, if the mean dimension is finite, then the supremum in Lindenstrauss-Tsukamoto's double variational principle can be taken over the set of ergodic measures. Additionally, the double variational principle holds for various other measure-theoretic $ε$-entropies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08051 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean dimension and rate-distortion function revisited Yang, Rui Dynamical Systems Information Theory Around the mean dimensions and rate-distortion functions, using some tools from local entropy theory this paper establishes the following main results: $(1)$ We prove that for non-ergodic measures associated with almost sure processes, the mean Rényi information dimension coincides with the information dimension rate. This answers a question posed by Gutman and Śpiewak (in Around the variational principle for metric mean dimension, \emph{Studia Math.} \textbf{261}(2021) 345-360). $(2)$ We introduce four types of rate-distortion entropies and establish their relation with Kolmogorov-Sinai entropy. $(3)$ We show that for systems with the marker property, if the mean dimension is finite, then the supremum in Lindenstrauss-Tsukamoto's double variational principle can be taken over the set of ergodic measures. Additionally, the double variational principle holds for various other measure-theoretic $ε$-entropies. |
| title | Mean dimension and rate-distortion function revisited |
| topic | Dynamical Systems Information Theory |
| url | https://arxiv.org/abs/2510.08051 |