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Spectral notions of aperiodic order

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  • Various spectral notions have been employed to grasp the structure and the long-range order of point sets, in particular non-periodic ones. In this article, we present them in a unified setting and explain the relations between them. For the sake of readability, we use Delone sets in Euclidean space as our main object class, and present generalisations in the form of further examples and remarks.

    Mathematics Subject Classification: 37B10, 52C23, 43A25.

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  • Figure 1.  A central patch of the eightfold symmetric Ammann-Beenker tiling of the plane, which can be generated by an inflation rule and is thus a self-similar tiling; see [6,Sec. 6.1] for details. The set of its vertex points is an example of a Meyer set, hence it is also an FLC Delone set. Moreover, it is a regular model set, as described in detail in [6,Ex. 7.8].

    Figure 2.  Illustration of a central patch of the diffraction measure of the Ammann-Beenker point set of Figure 1, which has pure point diffraction. A Bragg peak of intensity $I$ at $k\in\mathcal{B}$ is represented by a disc of an area that is proportional to $I$ and centred at $k$. Here, $\mathcal{B}$ is a scaled version of $\mathbb{Z}[{\rm{e}}^{\pi {\rm{i}}/4}]$, which is a group; compare [6,Sec. 9.4.2] for details. Although $\mathcal{B}$ is dense, the figure only shows Bragg peaks beyond a certain threshold. In particular, there are no extinctions in this case. At the same time, this measure is the diffraction measure of the Delone dynamical system defined by the (strictly ergodic) hull of the Ammann-Beenker point set, and $\mathcal{B}$ is its dynamical spectrum for the translation action of $\mathbb{R}^{2}$ on the hull.

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