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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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: We consider the disklikeness of the planar self-affine tile $ T$ generated by an integral expanding matrix $ A$ and a consecutive collinear digit set $ {\mathcal{D}}= \{0, v, 2v, \cdots, (\vert q\vert-1)v \}\subset {\Bbb{Z}}^2$. Let $ f(x)=x^{2}+ p x+ q$ be the characteristic polynomial of $ A$. We show that the tile $ T$ is disklike if and only if $ 2\vert p\vert\leq \vert q+2\vert$. Moreover, $ T$ is a hexagonal tile for all the cases except when $ p=0$, in which case $ T$ is a square tile. The proof depends on certain special devices to count the numbers of nodal points and neighbors of $ T$ and a criterion of Bandt and Wang (2001) on disklikeness.


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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
PII: S 0002-9947(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.