Disklikeness of planar self-affine tiles

King Shun LEUNG, Ka Sing LAU

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32 Citations (Scopus)

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 V = {0,v,2v,- " ,(|q| - 1)v) ⊂ ℤ². Let f(x) = x² +px+q be the characteristic polynomial of A. We show that the tile T is disklike if and only if 2|p| ≤ |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. Copyright © 2007 American Mathematical Society.
Original languageEnglish
Pages (from-to)3337-3355
JournalTransactions of the American Mathematical Society
Volume359
Issue number7
DOIs
Publication statusPublished - Jul 2007

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Citation

Leung, K.-S., & Lau, K.-S. (2007). Disklikeness of planar self-affine tiles. Transactions of the American Mathematical Society, 359(7), 3337-3355.

Keywords

  • Development of Subject Knowledge
  • Mathematics