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Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings

Research paper by Michael Baake, Uwe Grimm

Indexed on: 25 Jun '20Published on: 25 Jul '19Published in: arXiv - Mathematics - Metric Geometry



Abstract

Primitive inflation tilings of the real line with finitely many tiles of natural length and a Pisot-Vijayaraghavan unit as inflation factor are considered. We present an approach to the pure point part of their diffraction spectrum on the basis of a Fourier matrix cocycle in internal space. This cocycle leads to a transfer matrix equation and thus to a closed expression of matrix Riesz product type for the Fourier transforms of the windows for the covering model sets. In general, these windows are complicated Rauzy fractals and thus difficult to handle. Equivalently, this approach permits a construction of the (always continuous) eigenfunctions for the translation dynamical system induced by the inflation rule. We review and further develop the underlying theory, and apply it to the family of Pisa substitutions, with special emphasis on the Tribonacci case.