Local mean dimension theory for sofic group actions

TL;DR

This paper develops a local mean dimension theory for sofic group actions, introducing mean dimension pairs, and explores properties of factors, the mean dimension map, and the complexity of subshifts with positive mean dimension.

Contribution

It introduces the concept of mean dimension pairs and provides conditions for positive mean dimension in factors of sofic group actions, also analyzing the Borel nature of the mean dimension map.

Findings

Non-trivial factors of sofic group actions often have positive mean dimension.

The mean dimension map is Borel measurable.

The set of subshifts with completely positive mean dimension is a complete coanalytic set.

Abstract

Using a local perspective, we introduce \textit{mean dimension pairs} and give sufficient conditions of when every non-trivial factor of a continuous group action of a sofic group $G$ has positive mean dimension. In addition we show that the mean dimension map is Borel, and that the set of subshifts with completely positive mean dimension of $[0,1]^G$, the full $G$-shift on the interval, is a complete coanalytic set in the set of all subshifts (hence not Borel). Our results are new even when the acting group is $\Z$.