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ALGORITHM FOR DETERMINING PURE POINTEDNESS OF SELF-AFFINE TILINGS
 

Summary: ALGORITHM FOR DETERMINING PURE POINTEDNESS OF
SELF-AFFINE TILINGS
Shigeki Akiyama a and Jeong-Yup Lee b
a: Department of Mathematics, Faculty of Science, Niigata University,
8050 Ikarashi-2, Nishi-ku Niigata, Japan (zip: 950-2181)
akiyama@math.sc.niigata-u.ac.jp
b: KIAS 207-43, Cheongnyangni 2-dong, Dongdaemun-gu,
Seoul 130-722, Korea
jylee@kias.re.kr
Abstract. Overlap coincidence in a self-affine tiling in Rd
is equivalent to pure point
dynamical spectrum of the tiling dynamical system. We interpret the overlap coincidence
in the setting of substitution Delone set in Rd
and find an efficient algorithm to check
the pure point dynamical spectrum. This algorithm is easy to implement into a computer
program. We give the program and apply it to several examples. In the course of the proof
of the algorithm, we show a variant of the conjecture of UrbaŽnski (Solomyak [43]) on the
Hausdorff dimension of the boundaries of fractal tiles.
Keywords: Pure point spectrum, Self-affine tilings, Coincidence, Substitution Delone sets, Meyer sets,
Algorithm, Quasicrystals, Hausdorff dimension, Fractals.

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics