Reem Yassawi

Torsion-free S-adic shifts and their spectrum

S-adic shifts are symbolic shifts that generalise substitutional shifts. To generate a substitutional shift, one iterates the same substitution, generating a substitutional language and shift space. To generate an S-adic shift, one is given a directive sequence of substitutions and they are composed to generate an S-adic language and shift space. An S-adic shift is called constant-length if it is generated by a sequence of substitutions all of which are constant-length, i.e., for each substitution there is an l such that all letters are mapped to words of length l.

The property of recognisability is crucial for finding the discrete part of the spectrum of such shifts.
We call a sequence of constant-length substitutions 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 which generates the same shift space. 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 that Dekking introduced for constant length substitutions.
This is joint work with Alvaró Bustos-Gajardo and Neil Manibo.