Boundary of the Rauzy fractal sets in $\RR \times \CC$ generated by $P(x)=x^4-x^3-x^2-x-1$

Fabien Durand (1) , Ali Messaoudi (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 Department of Mathematics

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.

Mots clés: Rauzy fractals; tilings; automata; numeration systems

Domaines: Théorie des nombres [math.NT]; Combinatoire [math.CO]

Fabien Durand, Ali Messaoudi. Boundary of the Rauzy fractal sets in $\RR \times \CC$ generated by $P(x)=x^4-x^3-x^2-x-1$. 2008. ⟨hal-00293932v1⟩