Mean dimension theory in symbolic dynamics for finitely generated amenable groups

TL;DR

This paper explores the relationship between topological entropy and mean dimension in symbolic dynamics for finitely generated amenable groups, revealing proportionality results and extending to measure-theoretic dimensions.

Contribution

It establishes a precise connection between mean dimension, Hausdorff dimension, and entropy for actions of polynomial growth groups, including measure entropy and rate distortion dimensions.

Findings

Metric mean dimension equals topological entropy times subgroup growth rate.

Mean Hausdorff dimension matches the same proportionality.

Results extend to rate distortion dimension and measure entropy.

Abstract

In this paper, we mainly elucidate a close relationship between the topological entropy and mean dimension theory for actions of polynomial growth groups. We show that metric mean dimension and mean Hausdorff dimension of subshifts with respect to the lower rank subgroup are equal to its topological entropy multiplied by the growth rate of the subgroup. Meanwhile, we also prove the above result holds for the rate distortion dimension of subshifts with respect to the lower rank subgroup and measure entropy. Furthermore, some relevant examples are indicated.