Osaka Journal of Mathematics

Boundary of the Rauzy fractal sets in $\mathbb {R} \times \mathbb {C}$ generated by $P(x) = x^{4} - x^{3} - x^{2} - x - 1$

Fabien Durand and Ali Messaoudi

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Abstract

We study the boundary of the $3$-dimensional Rauzy fractal $\mathcal{E} \subset \mathbb{R} \times \mathbb{C}$ generated by the polynomial $P(x) = x^{4}-x^{3}-x^{2}-x-1$. The finite automaton characterizing the boundary of $\mathcal{E}$ is given explicitly. As a consequence we prove that the set $\mathcal{E}$ has $18$ neighboors where $6$ of them intersect the central tile $\mathcal{E}$ in a point. Our construction shows that the boundary is generated by an iterated function system starting with $2$ compact sets.

Article information

Source
Osaka J. Math., Volume 48, Number 2 (2011), 471-496.

Dates
First available in Project Euclid: 6 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1315318349

Mathematical Reviews number (MathSciNet)
MR2831982

Zentralblatt MATH identifier
1268.11039

Subjects
Primary: 11B85: Automata sequences
Secondary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension [See also 11N99, 28Dxx] 28A80: Fractals [See also 37Fxx] 52C22: Tilings in $n$ dimensions [See also 05B45, 51M20] 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx]

Citation

Durand, Fabien; Messaoudi, Ali. Boundary of the Rauzy fractal sets in $\mathbb {R} \times \mathbb {C}$ generated by $P(x) = x^{4} - x^{3} - x^{2} - x - 1$. Osaka J. Math. 48 (2011), no. 2, 471--496. https://projecteuclid.org/euclid.ojm/1315318349


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