Orbit equivalence of Cantor minimal systems and their continuous spectra

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.

References in corpus (1)

• Equal column sum and equal row sum dimension group realizations

Cited by in corpus (6)

• Symbolic factors of $\mathcal{S}$-adic subshifts of finite alphabet rank
• On Topological Rank of Factors of Cantor Minimal Systems
• Strong Orbit Equivalence and Superlinear Complexity
• On The Dimension Group of Unimodular S-Adic Subshifts
• On balance properties of hypercubic billiard words
• On symbolic factors of $\mathcal{S}$-adic subshifts of finite topological rank