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 Z dsystems 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.

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Abstract. The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitution subshifts measuretheoretical and continuous eigenvalues are the same. It is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems. We give partial answers to this question.

Abstract. We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues E(X, T ) of the minimal Cantor system (X, T ) is a subgroup of the intersection I(X, T ) of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated to (X, T ) is trivial, the quotient group I(X, T )/E(X, T ) is torsion free. We give examples with non trivial infinitesimal subgroups where this property fails. We also provide some realization results.

In this paper we show that for every congruent monotileable amenable group G and for every metrizable Choquet simplex K, there exists a minimal G-subshift, which is free on a full measure set, whose set of invariant probability measures is affine homeomorphic to K. If the group is virtually abelian, the subshift is free. Congruent monotileable amenable groups are a generalization of amenable residually finite groups. In particular, we show that this class contains all the infinite countable virtually nilpotent groups. This article is a generalization to congruent monotileable amenable groups of one of the principal results shown in [3] for residually finite groups.