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arXiv · 1611.08019

On a class of Rauzy fractals without the finiteness property

Gustavo Antonio Pavani
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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.

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Gustavo Antonio Pavani. 2016-11-23. On a class of Rauzy fractals without the finiteness property. https://arxiv.org/abs/1611.08019

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