Orbit Equivalence Rigidity of Equicontinuous Systems - Cortez-Munoz, M (Oxford University Press, 2016)
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.