A class of cubic Rauzy Fractals

Abstract

In this paper, we study arithmetical and topological properties for a class of Rauzy fractals ${\mathcal{R}}_a$ given by the polynomial $x^3-a{x}^2+x-1$ where $a\ge 2$ is an integer. In particular, we prove the number of neighbors of ${\mathcal{R}}_a$ in the periodic tiling is equal to $8$. We also give explicitly an automaton that generates the boundary of ${\mathcal{R}}_a$. As a consequence, we prove that ${\mathcal{R}}_2$ is homeomorphic to a topological disk.