Ghorbel Rim and Zouari Sourour Facult\'e des Sciences de Sfax, Facult\'e des Sciences de Sfax

Abstract : In \cite{rhs1}, it is proved that the lengths of periods occurring in the $\beta$-expansion of a rational series $r$ noted by $Per_{\beta}(r)$ depend only on the denominator of the reduced form of $r$ for quadratic Pisot unit series. In this paper, we will show first that every rational $r$ in the unit disk has strictly periodic $\beta$-expansion for Pisot or Salem unit basis under some condition. Second, for this basis, if $r =\frac{P}{Q} $ is written in reduced form with $|P| < |Q|$, we will generalize the curious property ``$Per_{\beta}(\frac{P}{Q})=Per_{\beta}(\frac{1}{Q})$".

Keywords : formal power series, $\beta$-expansion, Pisot series, Salem series