A Full Characterization of Bertrand Numeration Systems

Among all positional numeration systems, the widely studied Bertrand numeration systems are defined by a simple criterion in terms of their numeration languages. In 1989, Bertrand-Mathis characterized them via representations in a real base beta. However, the given condition turns out to be not necessary. Hence, the goal of this paper is to provide a correction of Bertrand-Mathis' result. The main difference arises when beta is a Parry number, in which case two associated Bertrand numeration systems are derived. Along the way, we define a non-canonical beta-shift and study its properties analogously to those of the usual canonical one.