Substitutions and Cantor real numeration systems

Abstract

We consider Cantor real numeration system as a frame in which every non-negative real number has a positional representation. The system is defined using a bi-infinite sequence \(B=(\beta_n)_{n\in\mathbb{Z}}\) of real numbers greater than one. We introduce the set of B-integers and code the sequence of gaps between consecutive B-integers by a symbolic sequence in general over the alphabet \(\mathbb{N}\). We show that this sequence is S-adic. We focus on alternate base systems, where the sequence B of bases is periodic, and characterize alternate bases B in which B-integers can be coded by using a symbolic sequence \(\bf{v}_{\it B}\) over a finite alphabet. With these so-called Parry alternate bases we associate some morphisms and show that \(\bf{v}_{\it B}\) is a fixed point of their composition. We then provide two classes of Parry alternate bases B generating sturmian fixed points. The paper generalizes results of Fabre and Burdík et al. obtained for the Rényi numerations systems, i.e., in the case when the Cantor base B is a constant sequence.