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

Baake M, Grimm U (2020)
Documenta Mathematica 25: 2303-2337.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Baake, MichaelUniBi; Grimm, Uwe
Abstract / Bemerkung
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.
Stichworte
Inflation tiling; Rauzy fractal; model set; mathematical diffraction; Fourier cocycle
Erscheinungsjahr
2020
Zeitschriftentitel
Documenta Mathematica
Band
25
Seite(n)
2303-2337
eISSN
1431-0643
Page URI
https://pub.uni-bielefeld.de/record/2952144

Zitieren

Baake M, Grimm U. Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings. Documenta Mathematica . 2020;25:2303-2337.
Baake, M., & Grimm, U. (2020). Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings. Documenta Mathematica , 25, 2303-2337. https://doi.org/10.25537/dm.2020v25.2303-2337
Baake, M., and Grimm, U. (2020). Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings. Documenta Mathematica 25, 2303-2337.
Baake, M., & Grimm, U., 2020. Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings. Documenta Mathematica , 25, p 2303-2337.
M. Baake and U. Grimm, “Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings”, Documenta Mathematica , vol. 25, 2020, pp. 2303-2337.
Baake, M., Grimm, U.: Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings. Documenta Mathematica . 25, 2303-2337 (2020).
Baake, Michael, and Grimm, Uwe. “Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings”. Documenta Mathematica 25 (2020): 2303-2337.

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