Rational Laurent series with purely periodic $\beta$-expansions

Abstract

The aim of this paper is to give families of Pisot and Salem elements $\beta$ in $\mathbb{F}_{q}((x^{-1}))$ with the curious property that the $\beta$-expansion of any rational series in the unit disk $D(0,1)$ is purely periodic. In contrast, the only known family of reals with the last property are quadratic Pisot numbers $\beta>1$ that satisfy $\beta^{2} = n\beta+1$ for some integer $n \geq 1$.