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arXiv · 2312.13767

Substitutions and Cantor real numeration systems

Emilie Charlier·Célia Cisternino·Zuzana Masáková·Edita Pelantová
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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 $\Beta=(\beta_n)_{n\in\Z}$ of real numbers greater than one. We introduce the set of $\Beta$-integers and code the sequence of gaps between consecutive $\Beta$-integers by a symbolic sequence in general over the alphabet $\N$. We show that this sequence is $S$-adic. We focus on alternate base systems, where the sequence $\Beta$ of bases is periodic and characterize alternate bases $\Beta$, in which $\Beta$-integers can be coded using a symbolic sequence $v_{\Beta}$ over a finite alphabet. With these so-called Parry alternate bases we associate some substitutions and show that $v_\Beta$ is a fixed point of their composition. The paper generalizes results of Fabre and Burd\'ik et al.\ obtained for the R\'enyi numerations systems, i.e., in the case when the Cantor base $\Beta$ is a constant sequence.

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Emilie Charlier, Célia Cisternino, Zuzana Masáková, Edita Pelantová. 2023-12-21. Substitutions and Cantor real numeration systems. https://arxiv.org/abs/2312.13767

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