Purely periodic expansions in systems with negative base

We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama’s result for positive Pisot unit base $\beta$, we find a sufficient condition so that there exist an interval $J$ containing the origin such that the $(-\beta )$-expansion of every rational number from $J$ is purely periodic. We focus on the case of quadratic bases and demonstrate the following difference between the negative and positive bases: It is known that the finiteness property ($\mathrm{F}\mathrm{i}\mathrm{n}(\beta )=\mathbb{Z}[\beta ]$) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that $\mathrm{F}\mathrm{i}\mathrm{n}(-\beta )=\mathbb{Z}[\beta ]$ is not necessary in the case of negative bases.