SUBSTITUTIONS OVER INFINITE ALPHABET GENERATING (−β)-INTEGERS

Abstract

We study positional numeration systems with negative base called (−β)-expansions in a more general setting than that of Ito and Sadahiro. We give an admissibility criterion for (−β)-expansions and discuss the properties of the set of (−β)-integers, denoted by ℤ−β. We give a description of distances between consecutive (−β)-integers and show that ℤ−β can be coded by an infinite word over an infinite alphabet, which is a fixed point of a non-erasing non-trivial morphism. We give a set of examples where ℤ−β is coded by an infinite word over a finite alphabet.