Eigenvalues and Strong Orbit Equivalence

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 (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 with (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. © Cambridge University Press, 2015.