Eigenvalues of finite rank Bratteli-Vershik dynamical systems

TL;DR

This paper investigates the conditions under which eigenvalues are continuous or measurable in finite rank minimal Cantor systems, revealing that continuous eigenvalues originate from the stable subspace and measurable eigenvalues are always rational.

Contribution

It provides new criteria for identifying continuous and measurable eigenvalues in finite rank Bratteli-Vershik systems, including examples and a focus on Toeplitz systems.

Findings

Continuous eigenvalues derive from the stable subspace.

Measurable eigenvalues can be irrational, but are always rational in Toeplitz systems.

Measurable eigenvalues do not necessarily come from the stable space.

Abstract

In this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated to the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition to be a measurable eigenvalue. Then we consider two families of examples. A first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally we study Toeplitz type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove measurable eigenvalues are always rational but not necessarily continuous.