Application of waist inequality to entropy and mean dimension

TL;DR

This paper explores how waist inequality constrains entropy and mean dimension in dynamical systems, revealing new insights into embeddings and conditional entropy.

Contribution

It applies waist inequality to dynamical systems, establishing conditions for positive conditional mean dimension and advancing understanding of topological entropy.

Findings

Maps with larger mean dimension have positive conditional metric mean dimension

Provides new perspectives on non-embeddability in Hilbert cube shifts

Connects geometric inequalities to entropy and mean dimension theory

Abstract

Waist inequality is a fundamental inequality in geometry and topology. We apply it to the study of entropy and mean dimension of dynamical systems. We consider equivariant continuous maps between dynamical systems and assume that the mean dimension of the domain is larger than the mean dimension of the target. We exhibit several situations for which the maps necessarily have positive conditional metric mean dimension. This study has interesting consequences to the theory of topological conditional entropy. In particular it sheds new light on a celebrated result of Lindenstrauss and Weiss about minimal dynamical systems non-embeddable in the shift on the Hilbert cube.