Disklikeness of planar self-affine tiles

Author:

Leung King-Shun,Lau Ka-Sing

Abstract

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

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference26 articles.

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