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 On the period of $\beta$-expansion of Pisot or Salem series over $\mathbb{F}_{q}((x^{-1}))$ Bull. Korean Math. Soc. 2015 Vol. 52, No. 4, 1047-1057 https://doi.org/10.4134/BKMS.2015.52.4.1047Published online July 31, 2015 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 MSC numbers : 11R06, 37B50 Downloads: Full-text PDF