Baake, Michael; Grimm, Uwe

Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings

Doc. Math. 25, 2303-2337 (2020)
DOI: 10.25537/dm.2020v25.2303-2337
Communicated by Stefan Teufel


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 continuously representable) eigenfunctions for the translation dynamical system induced by the inflation rule. We review and further develop the underlying theory, and illustrate it with the family of Pisa substitutions, with special emphasis on the classic Tribonacci case.

Mathematics Subject Classification

11K70, 42B10, 52C23, 37B10, 37F25, 28A80


inflation tiling, Rauzy fractal, model set, mathematical diffraction, Fourier cocycle


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Baake, Michael
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
Grimm, Uwe
School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom