Torsion-free $S$-adic shifts and their spectrum

TL;DR

This paper investigates torsion-free $S$-adic shifts generated by constant-length morphisms, establishing their quasi-recognizability and analyzing their spectral properties with refined invariants.

Contribution

It introduces the concept of torsion-free directive sequences, proves their quasi-recognizability, and provides a detailed spectral analysis using extended notions of height and column number.

Findings

Torsion-free sequences generate quasi-recognizable shifts.

Quasi-recognizability can be replaced by recognizability.

Spectral properties are characterized by extended invariants.

Abstract

In this work we study $S$-adic shifts generated by sequences of morphisms that are constant-length. We call a sequence of constant-length morphisms torsion-free if any prime divisor of one of the lengths is a divisor of infinitely many of the lengths. We show that torsion-free directive sequences generate shifts that enjoy the property of quasi-recognizability which can be used as a substitute for recognizability. Indeed quasi-recognizable directive sequences can be replaced by a recognizable directive sequence. With this, we give a finer description of the spectrum of shifts generated by torsion-free sequences defined on a sequence of alphabets of bounded size, in terms of extensions of the notions of height and column number. We illustrate our results throughout with examples that explain the subtleties that can arise.