Algorithm for determining pure pointedness of self-affine tilings

Abstract

Overlap coincidence in a self-affine tiling in ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^d$ 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 \cite{Solomyak:08}) on the Hausdorff dimension of the boundaries of fractal tiles.

21 pages, 3 figures

References in corpus (2)

• Pure Point Dynamical and Diffraction Spectra
• Diffraction and Palm measure of point processes