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

TL;DR

This paper analyzes the boundary of a 3D Rauzy fractal generated by a specific polynomial, providing an explicit automaton and revealing its structure as an iterated function system with 18 neighborhoods.

Contribution

It explicitly characterizes the boundary automaton of the Rauzy fractal and demonstrates its boundary is generated by an iterated function system starting from two compact sets.

Findings

The boundary automaton is explicitly constructed.

The Rauzy fractal has 18 neighborhoods with 6 intersecting the central tile at a point.

The boundary is generated by an iterated function system from 2 initial sets.

Abstract

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.