Purely Periodic Points of Complex Pisot Expansions

Abstract

Using a ``complex Pisot number'' $\lambda \in \mathbb{C}$ with $|\lambda|>1$, the numerical expansion $\sum_{j=-k}^{\infty}{a_j}/{\lambda^j}$ of a complex number, where each digit $a_j$ is chosen from some finite set $\Gamma$ of $\mathbb{Z}[\lambda]$, was established recently as an analogue of $\beta$-numeration system $\sum_{j=-k}^{\infty}{b_j}/{\beta^j}$ of a real number, where $b_j \in \{0, 1, \cdots, \lfloor\beta\rfloor\}$. In this paper, we give a necessary and sufficient condition for a complex number to have eventually or purely periodic complex Pisot expansion.