Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings. (English) Zbl 1462.11060
One can note authors’ 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 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.”
A survey of this paper is devoted to certain inflation tilings. The setting of inflation tilings of the real line are described. A number of auxiliary notions are recalled.
The main results are proven with explanations and auxiliary discussions. In addition, special attention is given to illustrative examples from the family of Pisa substitutions (including some based on cubic and quartic number fields).
Finally, several extensions of this investigation and applications of the present results are briefly described.
“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.”
A survey of this paper is devoted to certain inflation tilings. The setting of inflation tilings of the real line are described. A number of auxiliary notions are recalled.
The main results are proven with explanations and auxiliary discussions. In addition, special attention is given to illustrative examples from the family of Pisa substitutions (including some based on cubic and quartic number fields).
Finally, several extensions of this investigation and applications of the present results are briefly described.
Reviewer: Symon Serbenyuk (Kyïv)
MSC:
| 11K70 | Harmonic analysis and almost periodicity in probabilistic number theory |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
| 37B10 | Symbolic dynamics |
| 37F25 | Renormalization of holomorphic dynamical systems |
| 28A80 | Fractals |
Software:
OEISOnline Encyclopedia of Integer Sequences:
Discriminant of the polynomial x^n - x^(n-1) - ... - x - 1.References:
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