Abstract: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 $\epsilon$-entropies.
From: Rui Yang [view email]
[v1]
Thu, 9 Oct 2025 10:37:23 UTC (21 KB)
[v2]
Sat, 4 Jul 2026 11:25:09 UTC (93 KB)
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