Eigenvalues of minimal Cantor systems

Fabien Durand (1) , Alexander Frank , Alejandro Maass (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]

In this article we give necessary and sufficient conditions that a complex number must satisfy to be a continuous eigenvalue of a minimal Cantor system. Similarly, for minimal Cantor systems of finite rank, we provide necessary and sufficient conditions for having a measure theoretical eigen-value. These conditions are established from the combinatorial information of the Bratteli-Vershik representations of such systems. As an application, from any minimal Cantor system, we construct a strong orbit equivalent system without irrational eigenvalues which shares all measure theoretical eigenvalues with the original system. In a second application a minimal Cantor system is constructed satisfying the so-called maximal continuous eigenvalue group property.

Domaines: Systèmes dynamiques [math.DS]

Fabien Durand, Alexander Frank, Alejandro Maass. Eigenvalues of minimal Cantor systems. Journal of the European Mathematical Society, 2019, 21 (3), pp.727-775. ⟨10.4171/JEMS/849⟩. ⟨hal-02455347⟩