Orbit Equivalence Rigidity of Equicontinuous Systems

The paper is focused on the study of continuous orbit equivalence for minimal equicontinuous systems. We establish that every equicontinuous system is topologically conjugate to a profinite action, where the finite-index subgroups are not necessarily normal. We then show that two profinite actions $(X,G)$ and $(Y,H)$ are continuously orbit equivalent if and only if the groups $G$ and $H$ are virtually isomorphic, and the isomorphism preserves the structure of the finite-index subgroups defining the actions. As a corollary, we obtain a dynamical classification of the restricted isomorphism between generalized Bunce-Deddens $C^∗$-algebras. We show that, for minimal equicontinuous $\mathbb Z^d$-systems, continuous orbit equivalence implies that the systems are virtually piecewise conjugate. This result extends Boyle s flip-conjugacy theorem. We also show that the topological full group of a minimal equicontinuous system $(X,G)$ is amenable if and only if the group $G$ is amenable. © 2016 U.S. Government under licence to the London Mathematical Society, with the exception of the United States of America where no copiright protection exists.