On an analogue of the Hurewicz theorem for mean dimension

TL;DR

This paper investigates whether the classical Hurewicz theorem's analogue applies to mean dimension in dynamical systems, finding it generally does not, except when the base system has zero mean dimension.

Contribution

It demonstrates that the Hurewicz theorem analogue fails for mean dimension in general but holds under the condition of zero mean dimension in the base system.

Findings

The analogue of the Hurewicz theorem does not hold universally for mean dimension.

The analogue holds true when the base system has zero mean dimension.

Provides insights into the limitations and conditions for mean dimension analogues.

Abstract

The Hurewicz theorem is a fundamental result in classical dimension theory concerning continuous maps which lower topological dimension. We study whether or not its analogue holds for mean dimension of dynamical systems. Our first main result shows that an analogue of the Hurewicz theorem does not hold for mean dimension in general. Our second main result shows that it holds true if a base system has zero mean dimension.