Torsion-free S-adic shifts and their spectrum

27.01.2023 14:00 - 14:45

Reem Yassawi (Queen Mary University London)

Abstract: An S-adic shift is constant-length if it is generated by a sequence of morphisms all of which are constant-length, i.e., for each morphism there is an l such that all letters are mapped to words of length l. 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. This is joint work with Alvaró Bustos-Gajardo and Neil Manibo. 

H. Bruin

SR. 16, 3.OG, OMP 1