On the Fundamental Group of self-affine plane Tiles

Jun Luo[1]; Jörg M. Thuswaldner[2]

  • [1] Sun Yat-Sen University School of Mathematics and Computational Science Guangzhou 510275 (China)
  • [2] Institut für Mathematik und Angewandte Geometrie Abteilung für Mathematik und Statistik Montanuniversität Leoben Franz-Josef-Strasse 18 8700 Leoben (Austria)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 7, page 2493-2524
  • ISSN: 0373-0956

Abstract

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Let A ∈ ℤ 2 × ℤ 2 be an expanding matrix, 𝒟 ⊂ ℤ 2 a set with | det ( A ) | elements and define 𝒯 via the set equation A 𝒯 = 𝒯 + 𝒟 . If the two-dimensional Lebesgue measure of 𝒯 is positive we call 𝒯 a self-affine plane tile. In the present paper we are concerned with topological properties of 𝒯 . We show that the fundamental group π 1 ( 𝒯 ) of 𝒯 is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of π 1 ( 𝒯 ) . Furthermore, we give a short proof of the fact that the closure of each component of int ( 𝒯 ) is a locally connected continuum (we prove this result even in the more general case of plane IFS attractors fulfilling the open set condition). If π 1 ( 𝒯 ) = 0 we even show that the closure of each component of int ( 𝒯 ) is homeomorphic to a closed disk.We apply our results to several examples of tiles which are studied in the literature.

How to cite

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Luo, Jun, and Thuswaldner, Jörg M.. "On the Fundamental Group of self-affine plane Tiles." Annales de l’institut Fourier 56.7 (2006): 2493-2524. <http://eudml.org/doc/10211>.

@article{Luo2006,
abstract = {Let $A\in \mathbb\{Z\}^\{2\} \times \mathbb\{Z\}^2$ be an expanding matrix, $\mathcal\{D\}\subset \mathbb\{Z\}^2$ a set with $|\det (A)|$ elements and define $\{\mathcal\{T\}\}$ via the set equation $A\mathcal\{T\}=\mathcal\{T\}+\mathcal\{D\}$. If the two-dimensional Lebesgue measure of $\mathcal\{T\}$ is positive we call $\mathcal\{T\}$ a self-affine plane tile. In the present paper we are concerned with topological properties of $\mathcal\{T\}$. We show that the fundamental group $\pi _1(\mathcal\{T\})$ of $\{\mathcal\{T\}\}$ is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of $\pi _1(\mathcal\{T\})$. Furthermore, we give a short proof of the fact that the closure of each component of $\{\rm int\}(\{\mathcal\{T\}\})$ is a locally connected continuum (we prove this result even in the more general case of plane IFS attractors fulfilling the open set condition). If $\pi _1(\{\mathcal\{T\}\})=0$ we even show that the closure of each component of $\rm int(\mathcal\{T\})$ is homeomorphic to a closed disk.We apply our results to several examples of tiles which are studied in the literature.},
affiliation = {Sun Yat-Sen University School of Mathematics and Computational Science Guangzhou 510275 (China); Institut für Mathematik und Angewandte Geometrie Abteilung für Mathematik und Statistik Montanuniversität Leoben Franz-Josef-Strasse 18 8700 Leoben (Austria)},
author = {Luo, Jun, Thuswaldner, Jörg M.},
journal = {Annales de l’institut Fourier},
keywords = {Tile; tiling; fundamental group; number system; tile},
language = {eng},
number = {7},
pages = {2493-2524},
publisher = {Association des Annales de l’institut Fourier},
title = {On the Fundamental Group of self-affine plane Tiles},
url = {http://eudml.org/doc/10211},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Luo, Jun
AU - Thuswaldner, Jörg M.
TI - On the Fundamental Group of self-affine plane Tiles
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 7
SP - 2493
EP - 2524
AB - Let $A\in \mathbb{Z}^{2} \times \mathbb{Z}^2$ be an expanding matrix, $\mathcal{D}\subset \mathbb{Z}^2$ a set with $|\det (A)|$ elements and define ${\mathcal{T}}$ via the set equation $A\mathcal{T}=\mathcal{T}+\mathcal{D}$. If the two-dimensional Lebesgue measure of $\mathcal{T}$ is positive we call $\mathcal{T}$ a self-affine plane tile. In the present paper we are concerned with topological properties of $\mathcal{T}$. We show that the fundamental group $\pi _1(\mathcal{T})$ of ${\mathcal{T}}$ is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of $\pi _1(\mathcal{T})$. Furthermore, we give a short proof of the fact that the closure of each component of ${\rm int}({\mathcal{T}})$ is a locally connected continuum (we prove this result even in the more general case of plane IFS attractors fulfilling the open set condition). If $\pi _1({\mathcal{T}})=0$ we even show that the closure of each component of $\rm int(\mathcal{T})$ is homeomorphic to a closed disk.We apply our results to several examples of tiles which are studied in the literature.
LA - eng
KW - Tile; tiling; fundamental group; number system; tile
UR - http://eudml.org/doc/10211
ER -

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