Substitutions over infinite alphabet generating (-\beta)-integers

TL;DR

This paper studies (-\beta)-expansions, a type of negative base positional numeration system, providing admissibility criteria, analyzing the structure of (-\beta)-integers, and describing their coding via infinite words.

Contribution

It introduces a generalized admissibility criterion for (-\beta)-expansions and characterizes the structure and coding of the set of (-\beta)-integers.

Findings

Characterization of distances within (-\beta)-integers

Coding of (-\beta)-integers by infinite words over an infinite alphabet

Existence of a fixed point of a non-erasing morphism for the coding

Abstract

This contribution is devoted to the study of positional numeration systems with negative base introduced by Ito and Sadahiro in 2009, called (-\beta)-expansions. We give an admissibility criterion for more general case of (-\beta)-expansions and discuss the properties of the set of (-\beta)-integers. We give a description of distances within this set and show that this set can be coded by an infinite word over an infinite alphabet, which is a fixed point of a non-erasing non-trivial morphism.