Substitutions and Cantor real numeration systems

[en] 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 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 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 using a symbolic sequence v_B over a finite alphabet. With these so-called Parry alternate bases we associate some substitutions and show that v_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 B is a constant sequence.

Disciplines :

Mathematics

Author, co-author :

Charlier, Emilie ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes

Cisternino, Célia

Masáková, Zuzana

Pelantová, Edita

Language :

English

Title :

Substitutions and Cantor real numeration systems

Publication date :

2025

Journal title :

Acta Mathematica Hungarica

ISSN :

0236-5294

eISSN :

1588-2632

Publisher :

Springer, Netherlands

Volume :

176

Pages :

15-47

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBi :

since 28 April 2024

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