  It is well known that real numbers with a purely periodic decimal expansion
are the rationals having, when reduced, a denominator coprime with 10. The aim
of this paper is to extend this result to beta-expansions with a Pisot base
beta which is not necessarily a unit: we characterize real numbers having a
purely periodic expansion in such a base; this characterization is given in
terms of an explicit set, called generalized Rauzy fractal, which is shown to
be a graph-directed self-affine compact subset of non-zero measure which
belongs to the direct product of Euclidean and p-adic spaces.
