Algorithm for determining pure pointedness of self-affine tilings

2010/03/15 by Shigeki Akiyama, Akiyama, Shigeki, Jeong-Yup Lee +1
Materials Science · Mathematics · #37B50 #52C22 #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #math.DS #math.MG #msc:37B50 #msc:52C22

paper · pdf · doi:10.48550/arxiv.1003.2898

21 pages, 3 figures

arxiv created 2010/03/15 · openalex publication_date 2010/03/15 · arxiv updated 2010/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Overlap coincidence in a self-affine tiling in \R^d 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 \R^d 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 the proof of the algorithm, we show a variant of the conjecture of Urbański (Solomyak \citeSolomyak:08) on the Hausdorff dimension of the boundaries of fractal tiles.

Related

• The computation of overlap coincidence in Taylor-Socolar substitution tiling
• Neighbors of self-affine tiles and Rauzy Fractals
• Determining pure discrete spectrum for some self-affine tilings
• Overlapping substitutions and tilings
• Combinatorial topology of three-dimensional self-affine tiles