Quasi-recognizability and continuous eigenvalues of torsion-free S-adic systems

Abstract: We discuss combinatorial and dynamical descriptions of S-adic systems generated by sequences of constant-length morphisms between alphabets of bounded size. For this purpose, we introduce the notion of quasi-recognisability, a strictly weaker version of recognisability but which is indeed enough to reconstruct several classical arguments of the theory of constant-length substitutions in this more general context. Furthermore, we identify a large family of directive sequences, which we call "torsion-free", for which quasi-recognisability is obtained naturally, and can be improved to actual recognisability with relative ease.

Using these notions we give S-adic analogues of the notions of column number and height for substitutions, including dynamical and combinatorial interpretations of each, and give a general characterisation of the maximal equicontinuous factor of the identified family of S-adic shifts, showing as a consequence that in this context all continuous eigenvalues must be rational. As well, we employ the tools developed for a first approach to the measurable case.

This is a joint work with Neil Mañibo and Reem Yassawi.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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