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Relative topological entropy and relative mean dimension of induced factors

Kairan Liu, Yixiao Qiao

TL;DR

This paper investigates how relative entropy behaves under taking induced factors for amenable group actions. By leveraging local entropy theory and a novel combinatorial independence approach, the authors establish two sharp relations: (i) $h_{\mathrm{top}}(\pi,G)=0$ if and only if $h_{\mathrm{top}}(\widetilde{\pi},G)=0$, and (ii) $h_{\mathrm{top}}(\pi,G)>0$ implies $mdim(\widetilde{\pi},G)=+\infty$. The results connect fiber-wise entropy with mean dimension of the induced system, generalizing classical Glasner–Weiss and related findings to the relative and induced setting for amenable actions. The methods blend independence-density arguments with dimension-theoretic constructions, yielding a robust bridge between entropy zero/positive regimes and the infinite mean-dimension behavior of induced factors.

Abstract

We study the relation of relative topological entropy and relative mean dimension between a factor map and its induced factor map for amenable group actions. On the one hand, we prove that a factor map has zero relative topological entropy if and only if so does the induced factor map. On the other hand, we prove that a factor map has positive relative topological entropy if and only if the induced factor map has infinite relative mean dimension.

Relative topological entropy and relative mean dimension of induced factors

TL;DR

This paper investigates how relative entropy behaves under taking induced factors for amenable group actions. By leveraging local entropy theory and a novel combinatorial independence approach, the authors establish two sharp relations: (i) if and only if , and (ii) implies . The results connect fiber-wise entropy with mean dimension of the induced system, generalizing classical Glasner–Weiss and related findings to the relative and induced setting for amenable actions. The methods blend independence-density arguments with dimension-theoretic constructions, yielding a robust bridge between entropy zero/positive regimes and the infinite mean-dimension behavior of induced factors.

Abstract

We study the relation of relative topological entropy and relative mean dimension between a factor map and its induced factor map for amenable group actions. On the one hand, we prove that a factor map has zero relative topological entropy if and only if so does the induced factor map. On the other hand, we prove that a factor map has positive relative topological entropy if and only if the induced factor map has infinite relative mean dimension.

Paper Structure

This paper contains 9 sections, 10 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $G$-systems and amenable groups
  4. Induced systems and induced factors
  5. Relative topological entropy and independence
  6. Relative mean dimension
  7. Simplices and the Lebesgue lemma
  8. Proof of ${\text{\rm h}}_{\text{\rm top}}(\pi,G)=0\implies{\text{\rm h}}_{\text{\rm top}}(\widetilde{\pi},G)=0$
  9. Proof of ${\text{\rm h}}_{\text{\rm top}}(\pi,G)>0\implies{\rm mdim}(\widetilde{\pi},G)=+\infty$

Key Result

Theorem 1.1

Let $G$ be a countably infinite amenable group, $\pi:(X,G)\to(Y,G)$ a factor map between two $G$-systems, and $\widetilde{\pi}:({\mathcal{M}}(X),G)\to({\mathcal{M}}(Y),G)$ the induced factor map. Then we have:

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2: LW00
  • Proposition 2.3
  • proof
  • Theorem 2.4: cf. LW00
  • Lemma 2.5: BS25
  • Lemma 3.1: LW24
  • Lemma 3.2
  • proof
  • ...and 8 more