SUBSTITUTIONS OVER INFINITE ALPHABET GENERATING (-β)-INTEGERS

We study positional numeration systems with negative base called (-beta)-expansions in a more general setting than that of Ito and Sadahiro. We give an admissibility criterion for (-beta)-expansions and discuss the properties of the set of (-beta)-integers, denoted by Z_(beta). We give a description of distances between consecutive (-beta)-integers and show that Z_(beta) 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_(beta) is coded by an infinite word over a finite alphabet.