On a class of Rauzy fractals without the finiteness property

TL;DR

This paper investigates the topological and arithmetical properties of a specific class of Rauzy fractals, revealing boundary recognition via automata, neighbor counts, and conditions for non-disk topology, with detailed boundary structure analysis.

Contribution

It introduces explicit automata for boundary recognition of Rauzy fractals and characterizes their topological properties, including conditions for non-disk topology and boundary generation methods.

Findings

Boundary automata constructed for Rauzy fractals

Conditions under which fractals are not topological disks

Boundary of a specific fractal generated by iterated function systems

Abstract

We present some topological and arithmetical aspects of a class of Rauzy fractals $\mathcal{R}_{a,b}$ related to the polynomials of the form $P_{a,b}(x)=x^{3}-ax^{2}-bx-1$, where $a$ and $b$ are integers satisfying $-a+1 \leq b \leq -2$. This class has the property that $0$ lies on the boundary of $\mathcal{R}_{a,b}$. We construct explicit finite automata that recognize the boundaries of these fractals, which allows to establish the number of neighbors of $\mathcal{R}_{a,b}$. In particular, we prove that if $2a+3b+4 \leq 0$ then $\mathcal{R}_{a,b}$ is not homeomorphic to a topological disk. We also show that the boundary of the set $\mathcal{R}_{3,-2}$ is generated by two infinite iterated function systems.