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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Disklikeness of planar self-affine tiles
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by King-Shun Leung and Ka-Sing Lau PDF
Trans. Amer. Math. Soc. 359 (2007), 3337-3355 Request permission

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 ${\mathcal {D}}= \{0, v, 2v, \cdots , (|q|-1)v \}\subset {\Bbb {Z}}^2$. 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 $2|p|\leq |q+2|$. 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
  • MR Author ID: 190087
  • Email: kslau@math.cuhk.edu.hk
  • 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
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 3337-3355
  • MSC (2000): Primary 52C20, 52C22; Secondary 28A80
  • DOI: https://doi.org/10.1090/S0002-9947-07-04106-2
  • MathSciNet review: 2299458