Number representation using generalized (-β)-transformation

We study non-standard number systems with negative base -beta. Instead of the Ito-Sadahiro definition, based on the transformation T-beta of the interval [-beta/beta+1, 1/beta+1) into itself, we suggest a generalization using an interval [l, l + 1) with l is an element of (-1, 0]. Such numeration systems share many properties of positive base numeration introduced by Renyi, although the proofs are not always straightforward. In this paper we focus on the description of admissible digit strings and their periodicity. We address the question of the description of reference strings used in the admissibility condition. We give examples which contradict a result of Gora and show that in this aspect the negative base numeration significantly differs from the Renyi numeration. (C) 2011 Elsevier B.V. All rights reserved.