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 $\mathrm{B}=({\beta }_n{)}_{n\in \mathbb{Z}}$ of real numbers greater than one. We introduce the set of $\mathrm{B}$-integers and code the sequence of gaps between consecutive $\mathrm{B}$-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 $\mathrm{B}$ of bases is periodic and characterize alternate bases $\mathrm{B}$, in which $\mathrm{B}$-integers can be coded using a symbolic sequence $v_{\mathrm{B}}$ over a finite alphabet. With these so-called Parry alternate bases we associate some substitutions and show that $v_{\mathrm{B}}$ is a fixed point of their composition. 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 $\mathrm{B}$ is a constant sequence.