Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems

TL;DR

This paper establishes exact conditions for eigenvalues in linearly recurrent Cantor systems and provides an explicit example illustrating the complexity of their spectral structure.

Contribution

It offers necessary and sufficient criteria for eigenfunctions and constructs a specific example with nontrivial Kronecker and trivial maximal equicontinuous factors.

Findings

Derived precise conditions for measurable and continuous eigenfunctions.

Constructed an example with nontrivial Kronecker factor and trivial maximal equicontinuous factor.

Abstract

We give necessary and sufficient conditions to have measurable and continuous eigenfunctions for linearly recurrent Cantor dynamical systems. We also construct explicitly an example of linearly recurrent system with nontrivial Kronecker factor and a trivial maximal equicontinuous factor.