A new notion of dimension for dynamical systems and shift embeddability

TL;DR

This paper introduces a new dimension concept for dynamical systems that explains all known obstructions to shift embeddability, refuting a major conjecture.

Contribution

It proposes a novel dimension notion applicable to systems over any countable group, unifying and extending previous obstruction results.

Findings

The new dimension accounts for all known obstructions to shift embeddability.

Refutes a major conjecture in the field.

Provides a unified framework for understanding shift embeddability.

Abstract

A dynamical system $(X,T)$ is \emph{shift embeddable} if $(X,T)$ embeds continuously and equivariantly in the shift over $[0,1]^d$ for some finite $d$. Refuting a major conjecture in the field, in a recent result of Dranishnikov and Levin it was shown that Gromov's mean dimension and Lebesgue covering dimension of finite orbits are not the only obstructions for shift embeddability. We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embeddability.