A full characterization of Bertrand numeration systems

TL;DR

This paper corrects and extends Bertrand-Mathis' characterization of Bertrand numeration systems, especially for Parry numbers, by introducing a non-canonical beta-shift and analyzing its properties.

Contribution

It provides a corrected characterization of Bertrand numeration systems and introduces a non-canonical beta-shift for Parry numbers, enriching the understanding of these systems.

Findings

Corrected the characterization of Bertrand numeration systems.

Identified two associated systems for Parry numbers.

Analyzed properties of the non-canonical beta-shift.

Abstract

Among all positional numeration systems, the widely studied Bertrand numeration systems are defined by a simple criterion in terms of their numeration languages. In 1989, Bertrand-Mathis characterized them via representations in a real base $\beta$. However, the given condition turns to be not necessary. Hence, the goal of this paper is to provide a correction of Bertrand-Mathis' result. The main difference arises when $\beta$ is a Parry number, in which case are derived two associated Bertrand numeration systems. Along the way, we define a non-canonical $\beta$-shift and study its properties analogously to those of the usual canonical one.