Mean dimension and a non-embeddable example for amenable group actions

TL;DR

The paper constructs specific minimal actions of amenable groups with prescribed mean dimension that cannot be embedded into certain shift spaces, highlighting limitations in embedding theory for group actions.

Contribution

It introduces a method to construct minimal amenable group actions with arbitrary mean dimension that are non-embeddable into full shifts, revealing new obstructions in dynamical systems.

Findings

Existence of minimal G-actions with mean dimension d/2 for any infinite amenable group

Construction of non-embeddable examples into full G-shifts on ([0,1]^d)^G

Demonstration of limitations in embedding amenable group actions into shift spaces.

Abstract

For every infinite (countable discrete) amenable group $G$ and every positive integer $d$ we construct a minimal $G$-action of mean dimension $d/2$ which cannot be embedded in the full $G$-shift on $([0,1]^d)^G$.