Eigenvalues of minimal Cantor systems

TL;DR

This paper characterizes the complex eigenvalues of minimal Cantor systems using Bratteli-Vershik representations and constructs systems with specific eigenvalue properties.

Contribution

It provides necessary and sufficient conditions for continuous and measure-theoretical eigenvalues in minimal Cantor systems, and constructs systems with prescribed eigenvalue features.

Findings

Characterization of eigenvalues via Bratteli-Vershik data

Construction of systems without irrational eigenvalues

Examples of systems with maximal continuous eigenvalue group

Abstract

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 eigenvalue. 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.