Boundary of the Rauzy fractal sets in $\mathbb {R} \times \mathbb {C}$ generated by $P(x) = x^{4} - x^{3} - x^{2} - x - 1$

Abstract

We study the boundary of the $3$-dimensional Rauzy fractal $\mathcal{E} \subset \mathbb{R} \times \mathbb{C}$ generated by the polynomial $P(x) = x^{4}-x^{3}-x^{2}-x-1$. The finite automaton characterizing the boundary of $\mathcal{E}$ is given explicitly. As a consequence we prove that the set $\mathcal{E}$ has $18$ neighboors where $6$ of them intersect the central tile $\mathcal{E}$ in a point. Our construction shows that the boundary is generated by an iterated function system starting with $2$ compact sets.