A new notion of dimension for dynamical systems and shift embeddability

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.