Diffraction of a model set with complex windows

TL;DR

This paper analyzes the diffraction pattern of a ternary inflation tiling derived from the plastic number, using Fourier matrix cocycles to compute the diffraction measure of the associated model sets with Rauzy fractal windows.

Contribution

It introduces a method to explicitly compute the pure point diffraction measure of a complex model set using a Fourier matrix cocycle and a closed-form formula for the Fourier transform of Rauzy fractals.

Findings

Explicit formula for the diffraction measure obtained.

Rapid convergence of the infinite product for Fourier transform.

Connection between Rauzy fractals and diffraction patterns.

Abstract

The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals, via a rapidly converging infinite product.

Equations

$$M=\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 1\\ 0& 1& 0\end{array}\right),$$

$$\beta =\frac{(9+\sqrt{69}{)}^{\frac{1}{3}}+(9-\sqrt{69}{)}^{\frac{1}{3}}}{18^{\frac{1}{3}}}\approx 1.32472,$$

$$⟨u|=\frac{3+\beta +7{\beta }^2}{23}(1,\beta ,{\beta }^2)\text{and}|v⟩=(2-{\beta }^2,{\beta }^2-\beta ,\beta -1{)}^T,$$

$$\mathcal{B}=\left(\begin{array}{ccc}1& \beta & {\beta }^2\\ 1& \mathrm{Re}(\alpha )& \mathrm{Re}({\alpha }^2)\\ 0& \mathrm{Im}(\alpha )& \mathrm{Im}({\alpha }^2)\end{array}\right)\text{and}{\mathcal{B}}^{\ast }=\frac{2}{\sqrt{23}}\left(\begin{array}{ccc}\mathrm{Im}(\alpha ){\beta }^{-1}& \mathrm{Im}(\alpha )\beta & \mathrm{Im}(\alpha )\\ 2\mathrm{Im}(\alpha ){\beta }^2& -\mathrm{Im}(\alpha )\beta & -\mathrm{Im}(\alpha )\\ \beta & -1+\frac{3}{2}{\beta }^2& -\frac{3}{2}\beta \end{array}\right),$$

$$L^{⊛}=\frac{2\mathrm{Im}(\alpha )}{\beta \sqrt{23}}⟨1,{\beta }^2,\beta {⟩}_{\mathbb{Z}}=\frac{5-6\beta +4{\beta }^2}{23}L,$$

$$k\,=\,k(n_{1},n_{2},n_{3})\,=\,\frac{5-6\beta+4\beta^{2}}{23}\bigl{(}n_{0}+n_{1}\beta+n_{2}\beta^{2}\bigr{)}.$$

$$k^{\star}\,=\,\begin{pmatrix}\frac{1}{46}\bigl{(}(18n_{0}-4n_{1}+6n_{2})+(6n_{0}-9n_{1}+2n_{2})\beta-(4n_{0}-6n_{1}+9n_{2})\beta^{2}\bigr{)}\\ \frac{1}{2\sqrt{23}}\bigl{(}2n_{2}+(3n_{1}-2n_{0})\beta-3n_{2}\beta^{2}\bigr{)}\end{pmatrix}.$$

$$T=\left(\begin{array}{ccc}\varnothing & \varnothing & \{0\}\\ \{0\}& \varnothing & \{1\}\\ \varnothing & \{0\}& \varnothing \end{array}\right),$$

$${\Lambda }_a^{\prime }=\beta {\Lambda }_c,{\Lambda }_b^{\prime }=\beta {\Lambda }_a\cup (\beta {\Lambda }_c+1),{\Lambda }_c^{\prime }=\beta {\Lambda }_b.$$

$$W_a=\alpha W_c,W_b=\alpha W_a\cup (\alpha W_c+1),W_c=\alpha W_b,$$

$$\operatorname{Im}(\alpha)\bigl{(}\beta^{2}-1,\beta,1\bigr{)}\,=\,\tfrac{1_{\phantom{j}}}{2\sqrt{23}}\,\bigl{(}5-6\beta+4\beta^{2},-6-2\beta+9\beta^{2},4+9\beta-6\beta^{2}\bigr{)}.$$

$$\widehat{\gamma_{\omega}}\,=\sum_{k\in L^{\circledast}}I(k)\,\delta_{k}\quad\text{with}\quad I(k)\,=\,\Bigl{|}\sum_{i}h_{i}\,A_{i}(k)\Bigr{|}^{2},$$

$$A_i(k)=\frac{\mathrm{dens}({\Lambda }_i)}{\mathrm{vol}(W_i)}\text{rotatebox}180.0\hat{}{1}_{W_i}(k^{\star })=\frac{\mathrm{dens}(\Lambda )}{\mathrm{vol}(W)}\text{rotatebox}180.0\hat{}{1}_{W_i}(k^{\star }),$$

$$\text{rotatebox}180.0\hat{}{1}_{AK+t}(y)=|\det (A)|{\mathrm{e}}^{2\pi \mathrm{i}⟨t|y⟩}\text{rotatebox}180.0\hat{}{1}_K(A^{T}y),$$

$$\left(\begin{array}{c}{f}_a\\ f_b\\ f_c\end{array}\right)(y)={\beta }^{-1}\underline{B}(y)\left(\begin{array}{c}{f}_a\\ f_b\\ f_c\end{array}\right)(\underline{R}y),\text{with }\underline{R}=\underline{Q}{}^T\text{and }\underline{B}(y)=\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& {\mathrm{e}}^{2\pi \mathrm{i}{y}_1}\\ 0& 1& 0\end{array}\right).$$

$$C(0)=P=|v⟩⟨u|=\frac{3+\beta +7{\beta }^2}{23}\left(\begin{array}{ccc}2-{\beta }^2& \beta -1& {\beta }^2-\beta \\ {\beta }^2-\beta & 1+\beta -{\beta }^2& {\beta }^2-1\\ \beta -1& {\beta }^2-\beta & 1+\beta -{\beta }^2\end{array}\right).$$

$$A_i(k)=\mathrm{dens}(\Lambda )c_i(k^{\star })=\frac{3+\beta +7{\beta }^2}{23}{c}_i(k^{\star }),$$

Full text

Consider the primitive substitution $\varrho$ defined by $a\mapsto b\mapsto c\mapsto ab$ on the ternary alphabet $\{a,b,c\}$; see [4, Ex. 4.4] for details. Its substitution matrix reads

[TABLE]

with characteristic polynomial $x^{3}-x-1$. The latter is irreducible, with one real root,

[TABLE]

and a complex conjugate pair $\alpha,\overline{\alpha}$ of algebraic conjugates, where $\lvert\alpha\rvert^{2}=\beta^{-1}=\beta^{2}-1$. Below, we use $\alpha$ for the number with positive imaginary part.

The algebraic integer $\beta$ is a unit, and the smallest Pisot–Vijayaraghavan (PV) number, known as the plastic number; compare [4, p. 50 and Ex. 2.17]. It is also the Perron–Frobenius eigenvalue of $M$, where the corresponding left and right eigenvectors read

[TABLE]

which are strictly positive, and normalised such that $\langle 1|\hskip 0.5ptv\rangle=\langle u\hskip 0.5pt|\hskip 0.5ptv\rangle=1$. So, the entries of $|\hskip 0.5ptv\rangle$ are the relative frequencies of the letters in the symbolic sequences defined by $\varrho$, and $P\mathrel{\mathop{:}}=|\hskip 0.5ptv\rangle\langle u\hskip 0.5pt|$ is a projector of rank $1$, that is, $P^{2}=P$ and $\operatorname{im}(P)=\mathbb{R}\hskip 0.5pt\hskip 0.5pt|\hskip 0.5ptv\rangle$.

To turn the symbolic sequences into tilings, we choose intervals of natural length, namely $1,\beta$ and $\beta^{2}$, with control points on their left endpoints. For each sequence in the symbolic hull of $\varrho$, this leads to a typed point set, $\varLambda=\varLambda_{a}\hskip 0.5pt\hskip 0.5pt\dot{\cup}\hskip 0.5pt\hskip 0.5pt\varLambda_{b}\hskip 0.5pt\hskip 0.5pt\dot{\cup}\hskip 0.5pt\hskip 0.5pt\varLambda_{c}$, from the three types of control points. The average distance between neighbouring points in $\varLambda$ is $\bar{s}=4+2\beta-3\beta^{2}$, so we get $\operatorname{dens}(\varLambda)=1/\bar{s}=\frac{1}{23}(3+\beta+7\beta^{2})$ and $\operatorname{dens}(\varLambda_{i})=v_{i}\hskip 0.5pt\operatorname{dens}(\varLambda)$ for $i\in\{a,b,c\}$.

To continue, we work with the rank-$3$ $\mathbb{Z}$-module $L=\mathbb{Z}[\beta]=\langle 1,\beta,\beta^{2}\rangle_{\mathbb{Z}}$, which comprises all possible coordinates of our control points (relative to one of them, then considered as [math]). The Minkowski embedding [4, Ex. 3.5] of $L$ is a lattice $\mathcal{L}$ in $\mathbb{R}\hskip 0.5pt\!\times\!\mathbb{C}\simeq\mathbb{R}\hskip 0.5pt^{3}$, with $\operatorname{dens}(\mathcal{L})=2/\sqrt{23}$. A canonical choice for the basis matrix of $\mathcal{L}$ and its dual, $\mathcal{L}^{*}$, is

[TABLE]

where $\operatorname{Re}(\alpha)=-\beta/2$, $\operatorname{Re}(\alpha^{2})=1-\beta^{2}/2$, $\operatorname{Im}(\alpha)=\frac{4+9\beta-6\beta^{2}}{2\sqrt{23}}$ and $\operatorname{Im}(\alpha^{2})=\frac{6+2\beta-9\beta^{2}}{2\sqrt{23}}$ in our setting. We note in passing that duality can also be defined via the quadratic form (x,y)\mathrel{\mathop{:}}=xy+\operatorname{Re}\bigl{(}\sigma(x)\hskip 0.5pt\overline{\sigma(y)}\hskip 0.5pt\bigr{)}, where $\sigma\!:\,\mathbb{Q}(\beta)\longrightarrow\mathbb{Q}(\alpha)$ denotes the algebraic conjugation map defined by $\beta\mapsto\alpha$ and its unique extension to a field isomorphism.

We can extract the Fourier module from the first line of the dual basis matrix $\mathcal{B}^{*}$ as

[TABLE]

which is also the dynamical spectrum (in additive notation) of the tiling dynamical system as well as the (topologically equivalent) model set dynamical system defined by the control point sets; see [5] for background. Our system has pure point diffraction spectrum, because the defining point set is a regular model set (see below).

The Bragg peaks can be indexed by three integer Miller indices $(n_{0},n_{1},n_{2})$, where we parameterise the wave number, in line with (1), as

[TABLE]

The original version of the Minkowski embedding of $\mathbb{Z}[\beta]$ uses complex numbers, which is natural from an algebraic perspective. However, for Fourier analysis, we better work with real numbers, via $\mathbb{C}\simeq\mathbb{R}\hskip 0.5pt^{2}$, as initiated above for $\mathcal{B}$ and $\mathcal{B}^{*}$. Given the conjugation map $\sigma$, the $\star$-map: $\mathbb{Q}(\beta)\longrightarrow\mathbb{R}\hskip 0.5pt^{2}$ is conveniently defined by x\mapsto\bigl{(}\operatorname{Re}(\sigma(x)),\operatorname{Im}(\sigma(x))\bigr{)}^{T}. If $k\in L^{\circledast}$ is parameterised as in (2), this means

[TABLE]

Next, we need the displacement (or digit) matrix $T$ of our inflation rule in direct space; see [1] for definitions and properties. With the interval lengths as chosen above, it reads

[TABLE]

which is also reflected in the point set iteration

[TABLE]

It is known that we obtain a regular model set [9], with three topologically regular windows that have disjoint interiors. They are compact sets $W_{\!a},W_{b},W_{\!c}\subset\mathbb{C}$ that satisfy

[TABLE]

which defines a contractive iterated function system (IFS), so that the fixed point (and hence our window triple) is unique. Due to the nature of $\varLambda$ and $\varLambda_{i}$ as regular model sets, the volumes (areas) of the windows, compare [4, Ex. 7.6 and Fig. 7.3], satisfy the relations $\operatorname{dens}(\mathcal{L})\operatorname{vol}(W_{i})=v_{i}\hskip 0.5pt\operatorname{dens}(\varLambda)$, which results in the values

[TABLE]

The three windows are complex Rauzy fractals with a complicated topological structure [8]; see Figure 1 for an illustration and [7, Sec. 7.4] and references therein for background on Rauzy fractals. From the general diffraction result for regular model sets, see [4] and references therein, the weighted Dirac comb $\omega=\sum_{i\in\{a,b,c\}}h_{i}\,\delta_{\!\varLambda_{i}}$ with $h_{i}\in\mathbb{C}$ has diffraction

[TABLE]

because the Bombieri –Taylor property holds for primitive inflation rules [3]. Here, setting $W=W_{\!a}\cup W_{b}\cup W_{\!c}$, the amplitudes are given by

[TABLE]

where $1_{K}$ denotes the characteristic function of the set $K$. Though this formula looks nice, it is difficult to calculate \hskip 2.0pt\smash{\raisebox{16.5pt}{\rotatebox{180.0}{\widehat{\hskip 12.0pt}}}}\hskip-14.0pt{1_{W_{i}}} directly, due to the fractal nature of the windows. Let us next explain an alternative approach that harvests the inflation nature of our point sets.

We start by reformulating the window IFS (3) in $\mathbb{R}\hskip 0.5pt^{2}$. Recall that multiplication with $\alpha\in\mathbb{C}$ is matrix multiplication with $\hskip 0.5pt\underline{Q}\hskip 0.5pt=\left(\begin{smallmatrix}\operatorname{Re}(\alpha)&-\operatorname{Im}(\alpha)\\ \operatorname{Im}(\alpha)&\operatorname{Re}(\alpha)\end{smallmatrix}\right)$ in $\mathbb{R}\hskip 0.5pt^{2}$, where $\det(\hskip 0.5pt\underline{Q}\hskip 0.5pt)=\lvert\alpha\rvert^{2}=\beta^{-1}$. As the windows are interior-disjoint, Eq. (3), turns into equations in $L^{1}(\mathbb{R}\hskip 0.5pt^{2})$ for the characteristic functions of the windows and their images. Setting f_{i}(y)\mathrel{\mathop{:}}=\hskip 2.0pt\smash{\raisebox{16.5pt}{\rotatebox{180.0}{\widehat{\hskip 12.0pt}}}}\hskip-14.0pt{1_{W_{i}}}(y) and using the relation

[TABLE]

which holds for any real, invertible $2\!\times\!2$-matrix $A$ and compact set $K\subset\mathbb{R}\hskip 0.5pt^{2}$, one finds

[TABLE]

Here, $\hskip 0.5pt\underline{B\!}\,(y)$ is the internal Fourier matrix that emerges from the (inverse) Fourier transform of the $\star$-image of $T$, that is, $\hskip 0.5pt\underline{B\!}\,_{ij}(y)=\sum_{x\in T_{ij}}\hskip 0.5pt\mathrm{e}^{2\pi\mathrm{i}\hskip 0.5pt\langle x^{\star}|\hskip 0.5pty\rangle}$, where $y\in\mathbb{R}\hskip 0.5pt^{2}$.

Now, for $n\in\mathbb{N}$, we can define a cocycle via $\hskip 0.5pt\underline{B\!}\,^{(n)}(y)=\hskip 0.5pt\underline{B\!}\,(y)\hskip 0.5pt\underline{B\!}\,(\hskip 0.5pt\underline{R}\hskip 0.5pt\hskip 0.5pt\hskip 0.5pty)\cdots\hskip 0.5pt\underline{B\!}\,(\hskip 0.5pt\underline{R}\hskip 0.5pt\hskip 0.5pt^{n-1}y)$, with $\hskip 0.5pt\underline{B\!}\,^{(1)}=\hskip 0.5pt\underline{B\!}\,$ and $\hskip 0.5pt\underline{B\!}\,^{(n)}(0)=M^{n}$. Next, consider the matrix function $C(y)\mathrel{\mathop{:}}=\lim_{n\to\infty}\beta^{-n}\hskip 0.5pt\underline{B\!}\,^{(n)}(y)$, which is well defined because the limit exists for every $y\in\mathbb{R}\hskip 0.5pt^{2}$. In fact, $C$ is a continuous matrix function, and one has

[TABLE]

Moreover, one can show that $C(y)=|\hskip 0.5ptc(y)\rangle\langle u\hskip 0.5pt|$ with $|\hskip 0.5ptc(0)\rangle=|\hskip 0.5ptv\rangle$ and $|f(y)\rangle=\frac{\operatorname{dens}(\varLambda)}{\operatorname{dens}(\mathcal{L})}\,|\hskip 0.5ptc(y)\rangle$. In particular, one has $|\hskip 0.5ptc(y)\rangle=C(y)\hskip 0.5pt|\hskip 0.5ptv\rangle$, which makes the functions $f_{i}$ accessible.

For $k\in L^{\circledast}$ and $i\in\{a,b,c\}$, our amplitudes are

[TABLE]

and the corresponding intensities follow from Eq. (4). For any index triple, the intensity at the corresponding wave number $k\in L^{\circledast}$ is approximated by truncating the infinite product representation for $C(k^{\star})$ and calculating the amplitudes as explained above. Here, by using about 50–100 terms, one obtains the peaks with a relative precision better than $10^{-10}$.

Since the diffraction measure $\widehat{\gamma_{\omega}}$ can also be approximated (in the vague topology) by the absolute squares of exponential sums of finite approximants (divided by the system size), we illustrate the diffraction of the uniform Dirac comb (all $h_{i}=1$) in Figure 2 in comparison to an approximation for a finite system with $114$ tiles, obtained by $m=18$ inflation steps from an initial tile of type $a$. Here, as well as in Figure 3, the function $I_{m}(k)$ for the approximation refers to the (absolutely continuous) diffraction intensity for the corresponding finite system. The apparent mismatch emerges from the extremely slow convergence of the finite-size approximation, which should not come as a surprise due to the complicated fractal nature of the windows. To expand on the latter, we compare the neighbourhood of the peak with Miller indices $(1,2,2)$ at $k=(1+8\beta+10\beta^{2})/23\approx 1.267240014$ with approximations for various system sizes in Figure 3.

While the computation of the spectrum requires that $k\in L^{\circledast}$, the recursion equations (5) for the (inverse) Fourier transforms of the three windows can be used for any $y\in\mathbb{R}^{2}$. Thus, the (inverse) Fourier transform of the windows can be computed efficiently, despite the complex nature of the windows. As an example, the inverse Fourier transform f_{b}=\hskip 2.0pt\smash{\raisebox{16.5pt}{\rotatebox{180.0}{\widehat{\hskip 12.0pt}}}}\hskip-14.0pt{1_{W_{b}}} of the largest window is shown in Figure 4. In general, it is difficult to compute the Fourier transform of fractals; see, for instance, [6] for a related result on self-similar fractals of zero Lebesgue measure.

The cocycle method explained for this example works in full generality for all primitive PV inflation rules (with a unit inflation factor) that lead to regular model sets. In fact, it can even be applied to any primitive PV inflation rule, as well as to $S$-adic inflations with the same PV inflation multiplier and compatible tile sizes. When the spectrum is mixed, the cocycle method reproduces the pure point part; further details will be given in a forthcoming publication.

Acknowledgements

It is our pleasure to thank the MFO at Oberwolfach for hospitality, where this manuscript was completed. Our work was supported by the German Research Foundation (DFG), within the CRC 1283 at Bielefeld University, and by EPSRC through grant EP/S010335/1.