Ito-Sadahiro numbers vs. Parry numbers

TL;DR

This paper explores the algebraic properties of negative base numeration systems introduced by Ito and Sadahiro, focusing on Ito-Sadahiro numbers and their relation to Parry numbers, with implications for dynamical systems and number theory.

Contribution

It characterizes Ito-Sadahiro numbers in terms of their algebraic properties and compares them with Parry numbers within the context of negative base numeration systems.

Findings

Ito-Sadahiro numbers are characterized by the eventual periodicity of their $(-eta)$-expansion.

A comparison between Ito-Sadahiro numbers and Parry numbers reveals structural similarities and differences.

The paper establishes conditions under which the associated dynamical system is sofic.

Abstract

We consider positional numeration system with negative base, as introduced by Ito and Sadahiro. In particular, we focus on algebraic properties of negative bases $-\beta$ for which the corresponding dynamical system is sofic, which happens, according to Ito and Sadahiro, if and only if the $(-\beta)$-expansion of $-\frac{\beta}{\beta+1}$ is eventually periodic. We call such numbers $\beta$ Ito-Sadahiro numbers and we compare their properties with Parry numbers, occurring in the same context for R\'enyi positive base numeration system.