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Mean dimension and rate-distortion function revisited

Rui Yang

TL;DR

The work connects mean dimension with rate-distortion theory, proving that the mean Rényi information dimension coincides with the information-dimension rate for all invariant measures and introducing four rate-distortion entropies tied to Kolmogorov-Sinai entropy. It shows ergodic reductions for the Lindenstrauss-Tsukamoto double variational principle in finite mean-dimension systems with marker property and extends this principle to several measure-theoretic epsilon-entropies. By bridging local entropy notions with information-theoretic quantities, the paper offers a unified framework for variational principles of metric mean dimension and clarifies the interplay between dynamics and information theory in infinite-entropy contexts. These results enhance our understanding of embeddings, dimension theory, and entropy in dynamical systems, providing tools for analyzing both ergodic and non-ergodic behavior through rate-distortion and epsilon-entropy perspectives.

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 $ε$-entropies.

Mean dimension and rate-distortion function revisited

TL;DR

The work connects mean dimension with rate-distortion theory, proving that the mean Rényi information dimension coincides with the information-dimension rate for all invariant measures and introducing four rate-distortion entropies tied to Kolmogorov-Sinai entropy. It shows ergodic reductions for the Lindenstrauss-Tsukamoto double variational principle in finite mean-dimension systems with marker property and extends this principle to several measure-theoretic epsilon-entropies. By bridging local entropy notions with information-theoretic quantities, the paper offers a unified framework for variational principles of metric mean dimension and clarifies the interplay between dynamics and information theory in infinite-entropy contexts. These results enhance our understanding of embeddings, dimension theory, and entropy in dynamical systems, providing tools for analyzing both ergodic and non-ergodic behavior through rate-distortion and epsilon-entropy perspectives.

Abstract

Around the mean dimensions and rate-distortion functions, using some tools from local entropy theory this paper establishes the following main results: 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). We introduce four types of rate-distortion entropies and establish their relation with Kolmogorov-Sinai entropy. 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 -entropies.

Paper Structure

This paper contains 21 sections, 10 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mean dimension of phase space
  4. Metric mean dimension of phase space
  5. Measure-theoretic metric mean dimension of invariant measures
  6. Rate distortion theory
  7. Kolmogorov-Sinai $\epsilon$-entropy
  8. Brin-Katok's $\epsilon$-entropy
  9. Katok's $\epsilon$-entropies
  10. Pfister and Sullivan's $\epsilon$-entropy
  11. Proofs of main results
  12. An answer to Gutman-Śpiewak's open question
  13. Linking rate-distortion entropy and Kolmogorov-Sinai entropy
  14. $r$ rate-distortion entropy
  15. $L^p$ and $L^{\infty}$ rate-distortion entropies
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $([0,1]^{\mathbb{Z}}, \sigma)$ be a TDS with the metric $d^{\mathbb{Z}}$. Then for every $\mu\in M([0,1]^{\mathbb{Z}}, \sigma)$, where $d^{\mathbb{Z}}(x,y)=\sum_{n\in \mathbb{Z}}\frac{|x_n-y_n|}{2^{|n|}}$; ${\underline{\rm MRID}}([0,1]^{\mathbb{Z}},\sigma,d^{\mathbb{Z}},\mu)$ and ${\overline{\rm MRID}}([0,1]^{\mathbb{Z}},\sigma,d^{\mathbb{Z}},\mu)$ respectively denote the lower and upper mean

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Example 2.5
  • Lemma 3.1
  • ...and 15 more