Osaka Journal of Mathematics

On a class of Rauzy fractals without the finiteness property

Gustavo A. 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. This allows to establish the number of neighbors of $\mathcal{R}_{a,b}$ in the tiling it generates. Furthermore, 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.

Article information

Source
Osaka J. Math., Volume 56, Number 3 (2019), 577-599.

Dates
First available in Project Euclid: 16 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1563242425

Mathematical Reviews number (MathSciNet)
MR3982045

Zentralblatt MATH identifier
07108031

Subjects
Primary: 11B85: Automata sequences
Secondary: 28A80: Fractals [See also 37Fxx] 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx] 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]

Citation

Pavani, Gustavo A. On a class of Rauzy fractals without the finiteness property. Osaka J. Math. 56 (2019), no. 3, 577--599. https://projecteuclid.org/euclid.ojm/1563242425


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