Continuous eigenvalues of minimal subshifts via S-adic representations

TL;DR

This paper characterizes continuous eigenvalues in minimal symbolic dynamical systems with S-adic structures, linking combinatorics, coboundaries, and algebraic methods to understand spectral properties and balance conditions.

Contribution

It provides new characterizations of continuous eigenvalues for S-adic subshifts under mild conditions, connecting combinatorics, coboundaries, and linear algebra.

Findings

Characterizations of continuous eigenvalues using local coboundaries.

Conditions for nonexistence of trivial letter-coboundaries.

A simple criterion for letter-balance in primitive substitutive subshifts.

Abstract

We provide characterizations of continuous eigenvalues for minimal symbolic dynamical systems described by $S$-adic structures satisfying natural mild conditions, such as recognizability and primitiveness. Under the additional assumptions of finite alphabet rank or decisiveness of the directive sequence, these characterizations are expressed in terms of associated sequences of local coboundaries. We emphasize the role of combinatorics in the study of continuous eigenvalues through the interplay between coboundaries and extension graphs, and we give several types of sufficient conditions for the nonexistence of trivial letter-coboundaries. As further results, we apply coboundaries in the context of bounded discrepancy, and in particular we obtain a simple characterization of letter-balance for primitive substitutive subshifts. Moreover, we recover a result of Tijdeman on the minimal factor complexity of transitive subshifts with rationally independent letter frequencies. Finally, we use linear-algebraic duality to refine known descriptions of the possible values of eigenvalues in terms of measures of bases.