A class of cubic Rauzy Fractals

TL;DR

This paper investigates the properties of a class of Rauzy fractals defined by specific cubic polynomials, revealing their neighbor count, boundary automaton, and topological disk homeomorphism for a particular case.

Contribution

It introduces a detailed analysis of Rauzy fractals from cubic polynomials, including neighbor counts, boundary automata, and topological properties, advancing understanding of their structure.

Findings

Number of neighbors in the tiling is 8 for all ${\\mathcal R}_a$

Explicit automaton for boundary generation

${\mathcal R}_2$ is homeomorphic to a disk

Abstract

In this paper, we study arithmetical and topological properties for a class of Rauzy fractals ${\mathcal R}_a$ given by the polynomial $x^3- ax^2+x-1$ where $a \geq 2$ is an integer. In particular, we prove the number of neighbors of ${\mathcal R}_a$ in the periodic tiling is equal to $8$. We also give explicitly an automaton that generates the boundary of ${\mathcal R}_a$. As a consequence, we prove that ${\mathcal R}_2$ is homeomorphic to a topological disk.