Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Disk-like tiles and self-affine curves with noncollinear digits


Author: Ibrahim Kirat
Journal: Math. Comp. 79 (2010), 1019-1045
MSC (2000): Primary 52C20, 05B45; Secondary 37C70
DOI: https://doi.org/10.1090/S0025-5718-09-02301-1
Published electronically: September 24, 2009
MathSciNet review: 2600554
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A\in M_n(\mathbb{Z})$ be an expanding matrix, $ D\subset \mathbb{Z}^n$ a digit set and $ T=T(A,D)$ the associated self-affine set. It has been asked by Gröchenig and Haas (1994) that given any expanding matrix $ A\in M_2(\mathbb{Z})$, whether there exists a digit set such that $ T$ is a connected or disk-like (i.e., homeomorphic to the closed unit disk) tile. With regard to this question, collinear digit sets have been studied in the literature.

In this paper, we consider noncollinear digit sets and show the existence of a noncollinear digit set corresponding to each expanding matrix such that $ T$ is a connected tile. Moreover, for such digit sets, we give necessary and sufficient conditions for $ T$ to be a disk-like tile.


References [Enhancements On Off] (What's this?)

  • 1. S. Akiyama and J. M. Thuswaldner, Topological properties of two-dimensional number systems, J. Theor. Nombres Bordeaux, 12 (2000), 69-79. MR 1827838 (2002g:11013)
  • 2. C. Bandt and G. Gelbrich, Classification of self-affine lattice tilings, J. London Math. Soc., 50 (1994), 581-593. MR 1299459 (95g:52035)
  • 3. C. Bandt and Y. Wang, Disk-like self-affine tiles in $ \mathbb{R}^{2}$, Discrete Comput. Geom., 26 (2001), 591-601. MR 1863811 (2002h:52028)
  • 4. T. K. Dey, H. Edelsbrunner, and S. Guha, Computational topology in Advances In Discrete and Computational Geometry, B. Chazelle, J. E. Goodman and R. Pollack, eds., Contemp. Math., Vol. 223, American Mathematical Society, Providence, RI, 1999, pp. 109-143. MR 1661380 (2000a:68152)
  • 5. J. Gmainer and J. M. Thuswaldner, On disk-like self-affine tiles arising from polyominoes, Methods Appl. Anal. 13 (2006), no. 4, 351-371. MR 2384259 (2009g:28028)
  • 6. K. Gröchenig and A. Haas, Self-similar lattice tilings, J. Fourier Anal. Appl., 1 (1994), 131-170. MR 1348740 (96j:52037)
  • 7. M. Hata, On the structure of self-similar sets, Japan J. Appl. Math., 2 (1985), no. 2, pp. 381-414. MR 839336 (87g:58080)
  • 8. G.T. Herman and E. Zhao, Jordan surfaces in simply connected digital spaces, J. Math. Imaging Vision, 6 (1996), 121-138. MR 1390207 (97f:68199)
  • 9. I. Kirat and K.S. Lau, On the connectedness of self-affine tiles, J. London Math. Soc. 2, 62 (2000), 291-304. MR 1772188 (2001i:52027)
  • 10. I. Kirat, K.S. Lau and H. Rao, Expanding polynomials and connectedness of self-affine tiles, Discrete and Comput. Geometry, 31 (2004), 275-286. MR 2060641 (2005b:52052)
  • 11. J. C. Lagarias and Y. Wang, Integral self-affine tiles in $ {\mathbb{R}}^n$, Adv. Math., 121 (1996), 21-49. MR 1399601 (97d:52034)
  • 12. J. C. Lagarias and Y. Wang, Integral self-affine tiles in $ \mathbb{R}^{n}$. II. Lattice tilings, J. Fourier Anal. and Appl., 3 (1997), 84-102. MR 1428817 (98b:52026)
  • 13. K. S. Leung and K.S. Lau, Disklikeness of planar self-affine tiles, Trans. Amer. Math. Soc., 359 (2007), 3337-3355. MR 2299458 (2008k:52046)
  • 14. J. Luo, Boundary local connectivity of tiles in $ {\mathbb{R}}\sp 2$, Topology Appl., 154 (2007), no. 3, 614-618. MR 2280905 (2007j:54017)
  • 15. J. Luo, S. Akiyama and J. M. Thuswaldner, On the boundary connectedness of connected tiles, Math. Proc. Cambridge Philos. Soc., 137 (2004), no. 2, 397-410. MR 2092067 (2005g:37032)
  • 16. J. Luo, H. Rao and B. Tan, Topological structure of self-similar sets, Fractals, 10 (2002), 223-227. MR 1910665 (2003d:28014)
  • 17. S. M. Ngai and T. M. Tang. A technique in the topology of connected self-similar tiles, Fractals, 12 (2004), no.4, 389-403. MR 2109984 (2006b:52018)
  • 18. S. M. Ngai and T. M. Tang. Topology of connected self-similar tiles in the plane with disconnected interiors, Topology Appl., 150 (2005), no.1-3, 139-155. MR 2133675 (2006b:52019)
  • 19. A. Rosenfeld, Connectivity in digital pictures, J. Assoc. Comput. Mach., 17 (1970), 146-160. MR 0278576 (43:4306)
  • 20. K. Scheicher and J. M. Thuswaldner, Neighbors of self-affine tiles in lattice tilings, Fractals in Graz, 2001, Trends Math. Birkhäuser, Basel, 2003, pp. 241-262. MR 2091708 (2005f:37039)
  • 21. R. S. Strichartz and Y. Wang, Geometry of self-affine tiles. I. Indiana Univ. Math. J. (1) 48 (1999), 1-23. MR 1722192 (2000k:52017)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 52C20, 05B45, 37C70

Retrieve articles in all journals with MSC (2000): 52C20, 05B45, 37C70


Additional Information

Ibrahim Kirat
Affiliation: Department of Mathematics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey
Email: ibkst@yahoo.com

DOI: https://doi.org/10.1090/S0025-5718-09-02301-1
Keywords: Self-affine tiles, disk-like tiles, connectedness
Received by editor(s): November 7, 2007
Received by editor(s) in revised form: July 25, 2008
Published electronically: September 24, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society