A class of cubic Rauzy fractals

In this paper, we study arithmetical and topological properties for a class of Rauzy fractals RaRa given by the polynomial x3−ax2+x−1x3−ax2+x−1 where a≥2a≥2 is an integer. In particular, we prove the number of neighbors of RaRa in the periodic tiling is equal to 8. We also give explicitly an automaton that generates the boundary of RaRa. As a consequence, we prove that R2R2 is homeomorphic to a topological disk.