A new notion of dimension for dynamical systems and 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.

35 pages. Accepted for publication in GAFA (Geometric And Functional Analysis)