Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems

TL;DR

This paper investigates whether the known equivalence of measure-theoretical and continuous eigenvalues in substitution subshifts and odometers extends to the broader class of linearly recurrent Cantor systems, providing partial insights.

Contribution

It explores the relationship between measure-theoretical and continuous eigenvalues in linearly recurrent Cantor systems, offering partial results on their potential equivalence.

Findings

Measure-theoretical and continuous eigenvalues are the same for substitution subshifts and odometers.

Partial results suggest the equivalence may not hold universally for linearly recurrent Cantor systems.

Abstract

The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitution subshifts and odometers measure--theoretical and continuous eigenvalues are the same. It is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems. We give partial answers to this question.