We study the boundary of the 3-dimensional Rauzy fractal ${ \mathcal E} \subset \RR \times \CC$ 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 neighborhoods 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.
Categories: math.NT math.CO
