# 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

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topLuo, 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 -

## References

top- L. V. Ahlfors, Complex Analysis, (1966), McGraw-Hill Book Comp Zbl0154.31904MR510197
- S. Akiyama, J. M. Thuswaldner, A survey on topological properties of tiles related to number systems, Geom. Dedicata 109 (2004), 89-105 Zbl1073.37017MR2113188
- S. Akiyama, J. M. Thuswaldner, Topological structure of fractal tilings generated by quadratic number systems, Comput. Math. Appl. 49 (2005), 1439-1485 Zbl1123.11004MR2149493
- C. Bandt, Y. Wang, Disk-like self-affine tiles in ${\mathbb{R}}^{2}$, Discrete Comput. Geom. 26 (2001), 591-601 Zbl1020.52018MR1863811
- G. R. Conner, J. W. Lamoreaux, On the existence of universal covering spaces for metric spaces and subsets of the euclidean plane Zbl1092.57001MR2214874
- H. T. Croft, K. J. Falconer, R. K. Guy, Unsolved problems in geometry. Problem Books in Mathematics, (1994), Springer-Verlag, New York Zbl0748.52001MR1316393
- S. Eilenberg, Languages and Machines, A (1974), Academic Press, New York
- K. Falconer, Fractal Geometry, (1990), John Wiley and Sons Zbl0689.28003MR1102677
- K. Gröchenig, A. Haas, Self-similar lattice tilings, J. Fourier Anal. Appl. 1 (1994), 131-170 Zbl0978.28500MR1348740
- M. Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), 381-414 Zbl0608.28003MR839336
- A. Hatcher, Algebraic Topology, (2002), Cambridge University Press, Cambridge Zbl1044.55001MR1867354
- J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747 Zbl0598.28011MR625600
- I. Kátai, Number Systems and Fractal Geometry, (1995), Janus Pannonius University Pecs Zbl1029.11005
- A. Kovács, On the computation of attractors for invertible expanding linear operators in ${\mathbb{Z}}^{k}$, Publ. Math. Debrecen 56 (2000), 97-120 Zbl0999.11009MR1740496
- K. Kuratowski, Topology, I (1966), Academic Press, New York and London Zbl0158.40802MR217751
- K. Kuratowski, Topology, II (1968), Academic Press, New York and London Zbl0158.40802
- J. Luo, S. Akiyama, J. M. Thuswaldner, On the boundary connectedness of connected tiles, Math. Proc. Cambridge Philos. Soc. 137 (2004), 397-410 Zbl1070.37010MR2092067
- J. Luo, H. Rao, B. Tan, Topological structure of self-similar sets, Fractals 2 (2002), 223-227 Zbl1075.28005MR1910665
- E. E. Moise, Geometric Topology in Dimensions $2$ and $3$, 47 (1977), Springer-Verlag, New York-Heidelberg Zbl0349.57001MR488059
- S.-M. Ngai, T.-M. Tang, Vertices of self-similar tiles MR2210263
- S.-M. Ngai, T.-M. Tang, A technique in the topology of connected self-similar tiles, Fractals 12 (2004), 389-403 Zbl1304.28009MR2109984
- S.-M. Ngai, T.-M. Tang, Toplogy of self-similar tiles in the plane with disconnected interiors, Topology Appl. 150 (2005), 139-155 Zbl1077.37019MR2133675
- C. Pommerenke, Univalent Functions, (1975), Vandenhoeck & Ruprecht, Göttingen Zbl0298.30014MR507768
- K. Scheicher, J. M. Thuswaldner, Canonical number systems, counting automata and fractals, Math. Proc. Cambridge Philos. Soc. 133 (2002), 163-182 Zbl1001.68070MR1900260
- K. Scheicher, J. M. Thuswaldner, Neighbours of self-affine tiles in lattice tilings, Fractals in Graz 2001. Analysis, dynamics, geometry, stochastics (2003), 241-262, GrabnerP.P., Graz, Austria Zbl1040.52013MR2091708
- R. Strichartz, Y. Wang, Geometry of self-affine tiles I, Indiana Univ. Math. J. 23 (1999), 1-23 Zbl0938.52017MR1722192
- J. M. Thuswaldner, Attractors of invertible expanding linear operators and number systems in ${\mathbb{Z}}^{2}$, Publ. Math. Debrecen 58 (2001), 423-440 Zbl1012.11009MR1831051
- A. Vince, Digit tiling of euclidean space, Directions in Mathematical Quasicrystals (2000), 329-370, American Mathematical Society Providence, RI Zbl0972.52012MR1798999

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