Full Groups and Orbit Equivalence in Cantor Dynamics

In this note we consider dynamical systems $(X,G)$ on a Cantor set $X$ satisfying some mild technical conditions. The considered class includes, in particular, minimal and transitive aperiodic systems. We prove that two such systems $(X_1,G_1)$ and $(X_2,G_2)$ are orbit equivalent if and only if their full groups are isomorphic as abstract groups. This result is a topological version of the well-known Dye's theorem established originally for ergodic measure-preserving actions.