Purely Periodic beta-Expansions in the Pisot Non-unit Case
Résumé
It is well-known that real numbers with a purely periodic decimal expansion are the rationals having a denominator coprime with 10. We are interested in beta-expansions with a non-unit Pisot basis. We give a characterization of real numbers having a purely periodic expansion in such a basis. The characterization is given in terms of an explicit self-similar compact subset of non-zero measure in a product of a Euclidean space and p-adic spaces.