Disklikeness of planar self-affine tiles

Authors:
King-Shun Leung and Ka-Sing Lau

Journal:
Trans. Amer. Math. Soc. **359** (2007), 3337-3355

MSC (2000):
Primary 52C20, 52C22; Secondary 28A80

Published electronically:
February 13, 2007

MathSciNet review:
2299458

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the disklikeness of the planar self-affine tile generated by an integral expanding matrix and a consecutive collinear digit set . Let be the characteristic polynomial of . We show that the tile is disklike if and only if . Moreover, is a hexagonal tile for all the cases except when , in which case is a square tile. The proof depends on certain special devices to count the numbers of *nodal points* and *neighbors* of and a criterion of Bandt and Wang (2001) on disklikeness.

**[AG]**Shigeki Akiyama and Nertila Gjini,*On the connectedness of self-affine attractors*, Arch. Math. (Basel)**82**(2004), no. 2, 153–163. MR**2047669**, 10.1007/s00013-003-4820-z**[AT1]**Shigeki Akiyama and Jörg M. Thuswaldner,*A survey on topological properties of tiles related to number systems*, Geom. Dedicata**109**(2004), 89–105. MR**2113188**, 10.1007/s10711-004-1774-7**[AT2]**Shigeki Akiyama and Jörg M. Thuswaldner,*Topological properties of two-dimensional number systems*, J. Théor. Nombres Bordeaux**12**(2000), no. 1, 69–79 (English, with English and French summaries). MR**1827838****[AT3]**Shigeki Akiyama and J. M. Thuswaldner,*The topological structure of fractal tilings generated by quadratic number systems*, Comput. Math. Appl.**49**(2005), no. 9-10, 1439–1485. MR**2149493**, 10.1016/j.camwa.2004.09.008**[B]**Christoph Bandt,*Self-similar sets. V. Integer matrices and fractal tilings of 𝑅ⁿ*, Proc. Amer. Math. Soc.**112**(1991), no. 2, 549–562. MR**1036982**, 10.1090/S0002-9939-1991-1036982-1**[BG]**Christoph Bandt and Götz Gelbrich,*Classification of self-affine lattice tilings*, J. London Math. Soc. (2)**50**(1994), no. 3, 581–593. MR**1299459**, 10.1112/jlms/50.3.581**[BW]**C. Bandt and Y. Wang,*Disk-like self-affine tiles in ℝ²*, Discrete Comput. Geom.**26**(2001), no. 4, 591–601. MR**1863811**, 10.1007/s00454-001-0034-y**[Ba]**Michael F. Barnsley,*Fractals everywhere*, 2nd ed., Academic Press Professional, Boston, MA, 1993. Revised with the assistance of and with a foreword by Hawley Rising, III. MR**1231795****[Ga]**Adriano M. Garsia,*Arithmetic properties of Bernoulli convolutions*, Trans. Amer. Math. Soc.**102**(1962), 409–432. MR**0137961**, 10.1090/S0002-9947-1962-0137961-5**[Gi1]**William J. Gilbert,*Complex numbers with three radix expansions*, Canad. J. Math.**34**(1982), no. 6, 1335–1348. MR**678674**, 10.4153/CJM-1982-093-4**[Gi2]**William J. Gilbert,*Complex bases and fractal similarity*, Ann. Sci. Math. Québec**11**(1987), no. 1, 65–77 (English, with French summary). MR**912163****[HSV]**Derek Hacon, Nicolau C. Saldanha, and J. J. P. Veerman,*Remarks on self-affine tilings*, Experiment. Math.**3**(1994), no. 4, 317–327. MR**1341723****[H]**Masayoshi Hata,*On the structure of self-similar sets*, Japan J. Appl. Math.**2**(1985), no. 2, 381–414. MR**839336**, 10.1007/BF03167083**[KL]**Ibrahim Kirat and Ka-Sing Lau,*On the connectedness of self-affine tiles*, J. London Math. Soc. (2)**62**(2000), no. 1, 291–304. MR**1772188**, 10.1112/S002461070000106X**[KLR]**Ibrahim Kirat, Ka-Sing Lau, and Hui Rao,*Expanding polynomials and connectedness of self-affine tiles*, Discrete Comput. Geom.**31**(2004), no. 2, 275–286. MR**2060641**, 10.1007/s00454-003-2879-8**[LW1]**Jeffrey C. Lagarias and Yang Wang,*Integral self-affine tiles in 𝐑ⁿ. I. Standard and nonstandard digit sets*, J. London Math. Soc. (2)**54**(1996), no. 1, 161–179. MR**1395075**, 10.1112/jlms/54.1.161**[LW2]**Jeffrey C. Lagarias and Yang Wang,*Self-affine tiles in 𝑅ⁿ*, Adv. Math.**121**(1996), no. 1, 21–49. MR**1399601**, 10.1006/aima.1996.0045**[LW3]**Jeffrey C. Lagarias and Yang Wang,*Integral self-affine tiles in 𝑅ⁿ. II. Lattice tilings*, J. Fourier Anal. Appl.**3**(1997), no. 1, 83–102. MR**1428817**, 10.1007/s00041-001-4051-2**[L]**K.-S. Leung,*The radix expansions and the disklikeness of self-affine tiles*,. Ph.D. thesis, The Chinese University of Hong Kong, 2004.**[LAT]**Jun Luo, Shigeki Akiyama, and Jörg M. Thuswaldner,*On the boundary connectedness of connected tiles*, Math. Proc. Cambridge Philos. Soc.**137**(2004), no. 2, 397–410. MR**2092067**, 10.1017/S0305004104007625**[LRT]**Jun Luo, Hui Rao, and Bo Tan,*Topological structure of self-similar sets*, Fractals**10**(2002), no. 2, 223–227. MR**1910665**, 10.1142/S0218348X0200104X**[M]**D. Malone,*Solutions to dilation equations*, Ph.D. thesis, University of Dublin, 2000.**[NT]**Sze-Man Ngai and Tai-Man Tang,*A technique in the topology of connected self-similar tiles*, Fractals**12**(2004), no. 4, 389–403. MR**2109984**, 10.1142/S0218348X04002653**[O]**A. M. Odlyzko,*Nonnegative digit sets in positional number systems*, Proc. London Math. Soc. (3)**37**(1978), no. 2, 213–229. MR**507604**, 10.1112/plms/s3-37.2.213**[SK]**Hyun Jong Song and Byung Sik Kang,*Disclike lattice reptiles induced by exact polyominoes*, Fractals**7**(1999), no. 1, 9–22. MR**1687038**, 10.1142/S0218348X99000037**[T]**B. Tan,*Private communication*.

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Additional Information

**King-Shun Leung**

Affiliation:
Department of Mathematics, Science, Social Sciences and Technology, The Hong Kong Institute of Education, Tai Po, Hong Kong

Email:
ksleung@ied.edu.hk

**Ka-Sing Lau**

Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

Email:
kslau@math.cuhk.edu.hk

DOI:
http://dx.doi.org/10.1090/S0002-9947-07-04106-2

Keywords:
Digit sets,
neighbors,
nodal points,
radix expansion,
self-affine tiles

Received by editor(s):
October 14, 2004

Received by editor(s) in revised form:
June 23, 2005

Published electronically:
February 13, 2007

Additional Notes:
This research was partially supported by an HK RGC grant

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.