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January 2016 Topological properties of a class of cubic Rauzy fractals
Benoî t Loridant
Osaka J. Math. 53(1): 161-221 (January 2016).

Abstract

We consider the substitution $\sigma_{a, b}$ defined by \begin{align*} \sigma_{a, b}\colon & 1 \mapsto \underbrace{1\ldots 1}_{a}2{,} \\ & 2 \mapsto \underbrace{1\ldots 1}_{b}3{,} \\ & 3 \mapsto 1 \end{align*} with $a \geq b \geq 1$. The shift dynamical system induced by $\sigma_{a, b}$ is measure theoretically isomorphic to an exchange of three domains on a compact tile $\mathcal{T}_{a, b}$ with fractal boundary. We prove that $\mathcal{T}_{a, b}$ is homeomorphic to the closed disk iff $2b-a\leq 3$. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a Hölder continuous parametrization $C_{a, b}\colon \mathbb{S}^{1} \to \partial\mathcal{T}_{a, b}$ of the boundary of $\mathcal{T}_{a, b}$. As a by-product, this parametrization gives rise to an increasing sequence of polygonal approximations of $\partial\mathcal{T}_{a, b}$, whose vertices lye on $\partial\mathcal{T}_{a, b}$ and have algebraic pre-images in the parametrization.

Citation

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Benoî t Loridant. "Topological properties of a class of cubic Rauzy fractals." Osaka J. Math. 53 (1) 161 - 221, January 2016.

Information

Published: January 2016
First available in Project Euclid: 19 February 2016

zbMATH: 06546532
MathSciNet: MR3466830

Subjects:
Primary: 14C30
Secondary: 14D20 , 14L24 , 32J25

Rights: Copyright © 2016 Osaka University and Osaka City University, Departments of Mathematics

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Vol.53 • No. 1 • January 2016
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