Eigenvalues and strong orbit equivalence

TL;DR

This paper characterizes which subgroups of the circle can be realized as eigenvalues of minimal Cantor systems within a strong orbit equivalence class, linking algebraic properties of dimension groups to dynamical eigenvalues.

Contribution

It establishes conditions on subgroups of the circle for realization as eigenvalues and explores the structure of the additive group of eigenvalues in relation to the dimension group.

Findings

E(X,T) is a subgroup of I(X,T) for minimal Cantor systems.

When the infinitesimal subgroup is trivial, I(X,T)/E(X,T) is torsion free.

Examples show the property can fail with non-trivial infinitesimal subgroups.

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.