Continuous and Measurable Eigenfunctions of Linearly Recurrent Dynamical Cantor Systems

Abstract

Abstract. The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitution subshifts measuretheoretical 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.

Eigenvalues and strong orbit equivalence

“…This has been previously shown in [16], but has been also obtained in [5] (Proposition 11) without to be claimed. Proposition 11.…”

“…Dynamically speaking, it is a subgroup of I(X, T ) = ∩ µ∈M(X, T ) f dµ|f ∈ C(X, Z) , where M(X, T ) is the set of T -invariant probability measures of (X, T ) and C(X, Z) is the set of continuous functions from X to Z. An other proof of this observation can be found in [5] but it was not pointed out. In this paper we prove the following strong restriction.…”

“…For its proof we need to introduce some crucial quantities as it can be seen in the series of papers [5,3,4] and [7]. For any integer n ∈ N, we define the entrance time r n (x) of a point x ∈ X to the base B n by r n (x) = min{j ≥ 0; T j x ∈ B n }.…”

Necessary and Sufficient Conditions to Be an Eigenvalue for Linearly Recurrent Dynamical Cantor Systems

“…It is an extension of the concept of linearly recurrent subshift introduced in [4]. Most of the basic dynamical properties of linearly recurrent minimal Cantor systems are described in [1]. In particular, they are uniquely ergodic and the unique invariant measure is never strongly mixing.…”

Mathematics of Aperiodic Order

“…The authors of [CDHM,BDM1] generalize these results to the linearly recurrent symbolic systems and to finite rank systems in [BDM2]. An extension to Z d -action on a Cantor set is presented in [CGM].…”