Parametrization for a class of Rauzy Fractal

TL;DR

This paper investigates a family of Rauzy fractals defined by specific polynomials, providing explicit automata for their boundaries and proving their topological disk homeomorphism using an exotic numeration system.

Contribution

It introduces an explicit automaton for the boundary of Rauzy fractals and establishes their homeomorphism to a disk through a novel numeration approach.

Findings

Automaton explicitly describes Rauzy fractal boundaries.

Proves Rauzy fractals are topological disks.

Uses exotic numeration system for topological properties.

Abstract

In this paper, we study 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 give explicitly an automaton that generates the boundary of ${\mathcal R}_a$ and using an exotic numeration system we prove that ${\mathcal R}_a$ is homeomorphic to a topological disk.