Naive mean dimension

TL;DR

This paper explores the naive mean dimension in dynamical systems, establishing its bounds and relations with sofic mean dimension, especially for algebraic actions and nonamenable groups.

Contribution

It introduces the naive mean dimension concept, compares it with sofic mean dimension, and analyzes its properties for algebraic actions and nonamenable groups.

Findings

Naive mean dimension bounds sofic mean dimension for nonamenable groups.

For algebraic actions, naive mean rank provides better insights.

Relations between naive metric mean dimension and sofic metric mean dimension are established.

Abstract

We investigate the dynamical property of the naive mean dimension for continuous actions of any countable group on compact metrizable spaces. It is shown that naive mean dimension serves as an upper bound of sofic mean dimension for actions of nonamenable groups. For algebraic actions we obtain more satisfactory results by looking at the naive version of mean rank for modules over integral group rings. We also consider the naive metric mean dimension and investigate its relations with sofic metric mean dimension.