  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 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 are derived two associated Bertrand numeration systems.
Along the way, we define a non-canonical $\beta$-shift and study its properties
analogously to those of the usual canonical one.
