Orbit equivalence of Cantor minimal systems and their continuous spectra

TL;DR

This paper explores the relationship between continuous eigenvalues of Cantor minimal systems and their associated dimension groups, introducing the concept of irrational miscibility to characterize spectral properties.

Contribution

It introduces the concept of irrational miscibility of dimension groups and links it to the absence of non-trivial continuous eigenvalues in Cantor minimal systems.

Findings

Irrationally miscible dimension groups have no irrational continuous spectrum.

Strong orbit equivalence classes associated with such groups lack non-trivial continuous eigenvalues.

The paper characterizes spectral properties via the structure of dimension groups.

Abstract

To any continuous eigenvalue of a Cantor minimal system $(X,\,T)$, we associate an element of the dimension group $K^0(X,\,T)$ associated to $(X,\,T)$. We introduce and study the concept of irrational miscibility of a dimension group. The main property of these dimension groups is the absence of irrational values in the additive group of continuous spectrum of their realizations by Cantor minimal systems. The strong orbit equivalence (respectively orbit equivalence) class of a Cantor minimal system associated to an irrationally miscible dimension group $(G,\,u)$ (resp. with trivial infinitesimal subgroup) with trivial rational subgroup, have no non-trivial continuous eigenvalues.