Mean dimension and radius of comparison

Abstract

Consider a minimal and free topological dynamical system $(X, \Gamma)$, where $\Gamma$ is a discrete amenable group. Assume $(X, \Gamma)$ has the uniform Rokhlin property (URP) and the Cuntz comparison of open sets (COS), then it is shown that $$\mathrm{rc}(\mathrm{C}(X) \rtimes \Gamma) = \frac{1}{2} \mathrm{mdim}(X, \Gamma).$$ The same statement also holds for unital simple AH algebras with diagonal maps. The arguments in both cases are based on the recent preprint of OpenAI dealing with $\mathbb Z$-actions and the topological ideas used in that paper.