Weighted Mean Topological Dimension

TL;DR

This paper explores the properties of weighted mean topological dimension in dynamical systems, establishing its relationship with weighted metric mean dimension and weighted topological entropy, and identifying conditions for zero weighted mean dimension.

Contribution

It generalizes classical results by showing weighted mean dimension is not larger than weighted metric mean dimension and links it to weighted topological entropy.

Findings

Weighted mean dimension is bounded above by weighted metric mean dimension.

Systems with finite weighted topological entropy have zero weighted mean dimension.

Small boundary property implies zero weighted mean dimension.

Abstract

This paper is devoted to the investigation of the weighted mean topological dimension in dynamical systems. We show that the weighted mean dimension is not larger than the weighted metric mean dimension, which generalizes the classical result of Lindenstrauss and Weiss \cite{LWE}. We also establish the relationship between the weighted mean dimension and the weighted topological entropy of dynamical systems, which indicates that each system with finite weighted topological entropy or small boundary property has zero weighted mean dimension.