G ‐odometers and their almost one‐to‐one extensions

In this paper we recall the concepts of G ‐odometers and G ‐subodometers for G ‐actions, where G is a discrete finitely generated group; these generalize the notion of an odometer in the case G = ℤ. We characterize the G ‐regularly recurrent systems as the minimal almost one‐to‐one extensions of subodometers, from which we deduce that the family of the G ‐Toeplitz subshifts coincides with the family of the minimal symbolic almost one‐to‐one extensions of subodometers. We determine the continuous eigenvalues of these systems. When G is amenable and residually finite, a characterization of the G ‐invariant measures of these systems is given.