Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.08051 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of 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.