Alternate bases, alternate expansions and a generalization of Parry’s theorem

In this presentation, I state the results already obtained in the theory of alternate expansions. First of all, the well-known definitions and properties in the theory of beta-expansions are recalled. Then, I echo them when presenting their extensions in the case of the alternate bases. The famous Parry’s theorem is extended when considering the alternate base β1,β2,...,βp.