Coboundaries and eigenvalues of morphic subshifts

TL;DR

This paper introduces an efficient algorithm to compute eigenvalues of morphic subshifts using coboundaries, and characterizes their spectral properties, including conditions for weak mixing, expanding understanding of their dynamical behavior.

Contribution

It provides a novel algorithm for eigenvalue computation and characterizes the spectral structure of morphic subshifts, including conditions for weak mixing.

Findings

Efficient algorithm for eigenvalues using coboundaries

Continuous eigenvalues linked to coboundaries in S-adic subshifts

Necessary and sufficient conditions for weak mixing in unimodular cases

Abstract

We define a morphic subshift as a subshift generated by the image of a substitution subshift by another substitution. In other words, it is the subshift associated with a ultimately periodic directive sequence. We present an efficient algorithm for computing eigenvalues of morphic subshifts using coboundaries. We show that continuous eigenvalues of S-adic subshifts for primitive directive sequences are always associated with some coboundary. For morphic subshifts, we provide a characterization of the dimension of the Q-vector space generated by eigenvalues. Additionally, if the substitutions are unimodular and without coboundaries, we give a necessary and sufficient condition for weak mixing of the subshift.