Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
aa r X i v : . [ c ond - m a t . d i s - nn ] A p r Inflation versus projection sets:The role of the window in averaging and diffraction
Michael Baake , and Uwe Grimm , Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK School of Natural Sciences, University of Tasmania, Private Bag 37, Hobart TAS 7001, Australia
Abstract
Tilings based on the cut and project method are key model systems for the description of aperiodic solids. Typically,quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut and project systems,the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows.We illustrate this by the example of averaged shelling numbers for the Fibonacci tiling and review the standard approachto the diffraction for this example. Further, we discuss recent developments for inflation-symmetric cut and projectstructures, which are based on an internal counterpart of the renormalisation cocycle. Finally, we briefly review thenotion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.
Keywords:
Quasicrystals; projection method; inflation rules; averaging; autocorrelation; diffraction; hyperuniformity Introduction
The discovery of quasicrystals in the early 1980s[47] not only led to a reconsideration of the fun-damental concept of a crystal (see [32] and refer-ences therein), but also highlighted the need for amathematically robust treatment of the diffractionof systems that exhibit aperiodic order. The foun-dations for a rigorous approach were laid by Hof[33]. In particular, the measure-theoretic approachvia the autocorrelation and diffraction measures al-lows for a mathematically rigorous discussion andseparation of the different spectral components, thepure point, singular continuous and absolutely con-tinuous part; see [15] for background and examples,and [16, Sec. 9] for a systematic exposition. For gen-eral background on the theory of aperiodic order, werefer to [42, 3, 43, 16, 34, 1] and references therein.Within a few years, it was established that regu-lar model sets [38] (systems obtained by projectionfrom higher-dimensional lattices via cut and projectmechanisms with ‘nice’ windows) have pure pointdiffraction [46, 44]. We refer to the discussion in[16] for details and examples, and to [6] for an in-structive application of the cut and project approachto an experimentally observed structure with twelve-fold symmetry. The result on the pure point natureof diffraction holds for rather general setups, includ-ing cut and project schemes with non-Eucliden inter-nal spaces. It has recently been generalised to weakmodel sets of extremal densities [21, 45], for whichthe window may even entirely consist of boundary, that is, has no interior; see also [49, 50] for recentwork on pure point spectra.While systems based on a cut and project schemeare generally well understood, this is less so for sys-tems originating from substitution or inflation rules,which constitute another popular method of gener-ating systems with aperiodic order; see [16, 30] andreferences therein for details. There has been re-cent progress particularly on substitutions of con-stant length; see [36, 23, 24, 5, 11, 26].There are familiar examples of inflation-basedstructures for all spectral types, such as the Fi-bonacci chain for a pure-point diffractive system,the Thue–Morse chain for a system with purely sin-gular continuous diffraction, and the binary Rudin–Shapiro chain as the paradigm of a system with ab-solutely continuous diffraction; see [42, 3, 16] for de-tails. When one equips the Rudin–Shapiro chainwith balanced weights ( ± ± reformulated for point sets in aperiodic tilings, needboth a conceptual reformulation and new tools totackle them. The key observation is the necessity toemploy averaging concepts, and then tools from dy-namical systems and ergodic theory [43, 48, 13]. Ifone is in the favourable situation of point sets thatemerge from either the projection formalism or an in-flation procedure, many of these averaged quantitiesare well defined and can actually be calculated; see[13] and references therein. Despite some progress,many questions in this context remain open.Let us sketch how this introductory review is or-ganised. Our guiding example in this exposition isthe classic, self-similar Fibonacci tiling of the realline. Its descriptions as an inflation set and as a cutand project set are reviewed in Section 2. As a sim-ple example of the role of the window in averaging,we discuss the averaged shelling for this system inSection 3. This is followed by a brief review of thestandard approach to diffraction in Section 4, wherewe exploit the description of the Fibonacci point setas a cut and project set and the general results forthe diffraction of regular model sets.In Section 5, we introduce the recently developedinternal cocycle approach. For systems which pos-sess both an inflation and a projection interpreta-tion, such as the Fibonacci tiling, the inflation cocy-cle can be lifted to internal space. This makes it pos-sible to efficiently compute the diffraction of certaincut and project systems with complicated windows,such as windows with fractal boundaries which arecommonly found in inflation structures. We explorethis with planar examples, which are based on theFibonacci substitution, in Section 6.Finally, in Section 7, we discuss the use of ‘hy-peruniformity’ as a measure of order in Fibonaccisystems. This amounts to an investigation of the as-ymptotic behaviour of the total diffraction intensitynear the origin. It turns out that this can dinstin-guish between generic and inflation-invariant choicesfor the window in the cut and project scheme.2. The Fibonacci tiling revisited
Let us start with a paradigm of aperiodic order inone dimension, the classic Fibonacci tiling. It can bedefined via the primitive two-letter inflation rule ̺ : a ab, b a, where a and b represent tiles (or intervals) of length τ = (cid:0) √ (cid:1) and 1, respectively. The correspond-ing incidence matrix is given by(1) M = (cid:18) (cid:19) , which has Perron–Frobenius eigenvalue τ . Its leftand right eigenvectors read(2) h u | = τ + 25 (cid:0) τ, (cid:1) and | v i = (cid:0) τ − , τ − (cid:1) T , where we employ Dirac’s intuitive ‘bra-c-ket’ nota-tion, which makes it easy to distinguish row and col-umn vectors. We normalise the right eigenvector | v i such that h | v i = 1, which means that its entriesare the relative frequencies of the tiles. For laterconvenience, we normalise the left eigenvector h u | bysetting h u | v i = 1, rather than using the vector ofnatural tile lengths itself. With this normalisation,we have lim n →∞ τ − n M n = τ + 25 (cid:18) τ − τ − τ − (cid:19) = | v ih u | =: P , (3)where P = P is a symmetric projector of rank 1with spectrum { , } .Starting from the legal seed b | a , where the verti-cal bar denotes the origin, and iterating the square ofthe inflation rule ̺ , generates a tiling of the real linethat is invariant under ̺ ; see [16, Ex. 4.6] for detailsand why it does not matter which of the two fixedpoints of ̺ one chooses. Let us use the left end-points of each interval as control points and denotethe set of these points by Λ a and Λ b , respectively.Clearly, since 0 ∈ Λ a and all tiles have either length τ or length 1, all coordinates are integer linear com-binations of these two tile lengths, and we have Λ a,b ⊂ Z [ τ ] = { m + nτ : m, n ∈ Z } . While the incidence matrix M only contains informa-tion about the number of tiles under inflation, butnot about their positions, the latter information canbe encoded by introducing the displacement matrix (4) T = (cid:18) { } { }{ τ } ∅ (cid:19) , where ∅ denotes the empty set. Note that T isthe geometric counterpart of the instruction matri-ces that are used in the symbolic context [43]. Thematrix elements of T are sets that specify the rel-ative displacement for all tiles under inflation. Forinstance, the two entries in the first column corre-spond to a long tile with relative shift 0 and a smalltile with shift τ originating from inflating a long tile.Clearly, the inflation matrix M is recovered if onetakes the elementwise cardinality of T , noting thatthe empty set has cardinality 0.The inflation rule ̺ induces an iteration on pairsof point sets, namely Λ ( n +1) a = τ Λ ( n ) a ∪ τ Λ ( n ) b ,Λ ( n +1) b = τ Λ ( n ) a + τ , (5) b a a b a b a a b a b a L Figure 1.
Cut and project description of theFibonacci chain from the lattice L (blue dots).The windows W a and W b are the cross-sectionsof the yellow and green strips, repectively.with suitable initial conditions Λ (0) a,b . When one startswith the left endpoints of a legal seed, this iterationprecisely reproduces the endpoints of the correspond-ing, successive inflation steps. In this case, the unionon the right-hand side is disjoint. In particular, forthe above choice of Λ a,b , one needs Λ (0) a = { } and Λ (0) b = {− } .The point sets Λ a,b also have an interpretationas a cut and project set. Here, we use the natural(Minkowski) embedding of the module Z [ τ ] in theplane R , by associating to each x = m + nτ ∈ Z [ τ ]its image x ⋆ = m + nτ ⋆ = m + n (1 − τ ) under algebraicconjugation (which maps √ −√ L = (cid:8) ( x, x ⋆ ) : x ∈ Z [ τ ] (cid:9) = (cid:8) ( m + nτ, m + nτ ⋆ ) : m, n ∈ Z (cid:9) = (cid:8) m (1 ,
1) + n ( τ, τ ⋆ ) : m, n ∈ Z (cid:9) , which is a planar lattice with basis vectors (1 ,
1) and( τ, τ ⋆ ); see [16, 6] for details and further examples.Here, we refer to the two one-dimensional subspacesof R = R × R as the physical and the internal space,respectively. The physical space hosts our point sets Λ a,b , while the windows are subsets of the internalspace, with the ⋆ -map providing the relevant linkbetween the two spaces.The point sets Λ a,b are given by the projectionof two strips of the lattice L ; compare Figure 1.The strips are defined by their cross-sections, usu-ally called windows , which are the half-open inter-vals W a = [ τ − , τ −
1) and W b = [ − , τ − L = Z [ τ ], the projection of L into physical space, thepoint sets are thus given by(6) Λ a,b = (cid:8) x ∈ L : x ⋆ ∈ W a,b (cid:9) . The windows W a,b , or more precisely their closures,can be obtained as the unique solutions of a con-tractive iterated function system that arises from the ⋆ -image of (5),(7) W a = σW a ∪ σW b , W b = σW a + σ, where σ = τ ⋆ = 1 − τ satisfies | σ | <
1. One keyproperty, which can be employed to show that thesepoint sets are pure point diffractive, is the fact thatthe ⋆ -images of the point sets Λ a,b are uniformly dis-tributed in the windows W a,b , which makes it possibleto translate the computation of average quantities inphysical space to computations in internal space.3. Shelling
Let us discuss a simple example of an averagedquantity, the averaged shelling function for the Fi-bonacci point set; see [13] for the concept and var-ious applications to aperiodic systems. For a pointset, the shelling problem asks for the number n ( r )of points that lie on shells of radius r , taken withrespect to a fixed centre. For an aperiodic point set,this generally depends on the choice of the centre.The averaged shelling numbers a ( r ) are obtained bytaking the average over all choices of centres, wherewe limit ourselves to centres that are themselves inthe point set. Clearly, since we are dealing with aone-dimensional point set, any shell can have at mosttwo points, so n ( r ) ∈ { , , } for all r ∈ R , with n ( r ) = 0 if r Z [ τ ], as well as n (0) = a (0) = 1.Clearly, this also implies that a ( r ) ∈ [0 ,
2] for all r ∈ R , with a ( r ) = 0 whenever r Z [ τ ].Consider a point x ∈ Λ and r = m + nτ ∈ Z [ τ ]. Tocompute n ( r ), we have to check whether x ± r are alsoin the point set Λ . From the model set description,we know that x ⋆ ∈ W , and checking whether x ± r arein Λ is equivalent to checking whether x ⋆ ± r ⋆ ∈ W .While it is possible to perform this computation forany given value of x and r , there is no simple closedformula for these coefficients.To obtain the averaged shelling number, we haveto consider all x ∈ Λ as centres, each with the sameweight, which means averaging over all x ⋆ ∈ W . De-fine ν ( r ) = ν ( − r ) as the relative frequency to find apoint in Λ at x as well as at x + r , so a (0) = ν (0) = 1and a ( r ) = 2 ν ( r ) for r >
0, to account for the pointson both sides. Now, for r ∈ Z [ τ ], the frequency ν ( r )of having both x ⋆ ∈ W and x ⋆ + r ⋆ ∈ W can be cal-culated as the overlap length between the window W and the shifted window W − r ⋆ , divided by the lengthof W , which is | W | = τ . This is correct because theuniform distribution of points in the window [16, 39]implies that the frequency of any configuration isproportional to the length of the corresponding sub-window. Clearly, the length of the overlap betweenthese two intervals is 0 whenever | r ⋆ | > τ , and de-creases linearly with | r ⋆ | , so we get ν ( r ) = (cid:12)(cid:12) W ∩ ( W − r ⋆ ) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:12)(cid:12) = ( − | r ⋆ | τ , if r ∈ Z [ τ ] and | r ⋆ | τ ,0 , otherwise.(8) . . . . . ra ( r ) . . . . . . . . . . . . . . r ⋆ a ( r ) Figure 2.
Averaged shelling numbers a ( r ) forthe Fibonacci point set as a function of r (left)and r ⋆ (right).Consequently, the averaged shelling numbers for theFibonacci point set are given by a ( r ) = , if r = 0,2 (cid:0) − | r ⋆ | τ (cid:1) , if r ∈ Z [ τ ] with | r ⋆ | τ ,0 , otherwise . Note that a ( r ), for r ∈ Z [ τ ], is a simple function of r ⋆ , but that it behaves rather erratically if one looksat it as a function of r ; compare Figure 2. The reasonbehind this observation is the total discontinuity ofthe ⋆ -map from physical to internal space.For the one-dimensional example at hand, thenumbers ν ( r ) are nothing but the relative probability to find two points at a distance r , and thus the (rela-tively normalised) autocorrelation coefficients of thepoint set Λ . As such, they are intimately connectedto the diffraction of this point set. Clearly, corre-lations are much easier handled in internal space,where we can calculate them via volumes of intersec-tions of windows, as we shall see shortly.4. Standard approach to diffraction
Here, we start with a brief summary of the deriva-tion of the diffraction spectrum for the Fibonaccipoint set Λ = Λ a ∪ Λ b , considered as a cut and projectset Λ = { x ∈ L : x ⋆ ∈ W } with W = W a ∪ W b . As-sume that we place point scatterers of unit scatteringstrength at all points x ∈ Λ , and consider the corre-sponding Dirac comb ω = δ Λ := X x ∈ Λ δ x . We associate to ω the autocorrelation γ = ω ⊛ e ω ,where e ω is the ‘flipped-over’ (reflected) version of ω and ⊛ denotes volume-averaged (or Eberlein) convo-lution [16, Sec. 8.8]. The diffraction measure b γ is theFourier transform of the autocorrelation.From the general diffraction theory for cut andproject sets with well-behaved windows, we knowthat the diffraction measure of this system is apure point measure, so consists of Bragg peaks only.These Bragg peaks are located on the projection ofthe entire dual lattice L ∗ = 1 √ (cid:8) m ( τ − , τ ) + n (1 , −
1) : m, n ∈ Z (cid:9) to the physical space (the first coordinate), which is L ⊛ = √ Z [ τ ]. We call this set the Fourier module of the Fibonacci point set; it coincides with the dy-namical spectrum (in additive notation) in the math-ematical literature. Note that √ = (2 τ − /
5, so L ⊛ ⊂ Q ( τ ), which means that the ⋆ -map is well de-fined for all k ∈ L ⊛ . The Fourier module is a densesubset of R , which means that the diffraction con-sists of Bragg peaks on a dense set in space, wherethe intensities are locally summable.The diffraction measure is thus the countable sum b γ = X k ∈ L ⊛ | A ( k ) | δ k where the diffraction amplitudes, or Fourier–Bohr (FB) coefficients, are given by the general formula(9) A ( k ) = dens( Λ )vol( W ) c W ( − k ⋆ ) = dens( Λ )vol( W ) | W ( k ⋆ )for all k ∈ L ⊛ , and vanish otherwise. Here, 1 W de-notes the characteristic function of the window W ,defined by 1 W ( x ) = ( , if x ∈ W ,0 , otherwise,and b g and q g denote the Fourier transform and inverseFourier transform of an L -function g , respectively.With dens( Λ ) = ( τ + 2) / W ) = τ , Eq. (9)evaluates to A ( k ) = 1 √ Z τ − − e π i k ⋆ y d y = τ √ π i k ⋆ ( τ − sinc( πτ k ⋆ )where sinc( x ) = sin( x ) /x . Hence, the diffraction in-tensities are(10) I ( k ) = | A ( k ) | = (cid:18) τ √ πτ k ⋆ ) (cid:19) for all k ∈ L ⊛ , and 0 otherwise. This is illustratedin Figure 3.The corresponding autocorrelation measure γ canbe expressed in terms of the (dimensionless) pair cor-relation coefficients ν ( r ) := dens (cid:0) Λ ∩ ( Λ − r ) (cid:1) dens( Λ ) = ν ( − r ) , which are positive for all r ∈ Λ − Λ ⊂ Z [ τ ] and vanishfor all other distances r . These are precisely the coef-ficients we defined in Eq. (8) to compute the shellingnumbers. The link between the two expressions isprovided by the ⋆ -map and the uniform distributionof Λ ⋆ in the window W . In terms of these pair cor-relation coefficients, the autocorrelation measure is γ = dens( Λ ) X r ∈ Λ − Λ ν ( r ) δ r , Figure 3.
Schematic construction of thediffraction measure of the Fibonacci point setfrom the dual lattice L ∗ (blue dots). A point( k, k ⋆ ) ∈ L ∗ results in a Bragg peak at k ∈ L ⊛ of intensity given by the value of the functionon the right-hand side evaluated at k ⋆ .which is a pure point measure supported on the dif-ference set Λ − Λ .More generally, we may associate two different, ingeneral complex, scattering strengths u a and u b tothe points in Λ a and Λ b , respectively, and considerthe weighted Dirac comb ω = u a δ Λ a + u b δ Λ b . In thiscase, the diffraction intensity for all wave numbers k ∈ L ⊛ is given by the superposition(11) I ( k ) = (cid:12)(cid:12) u a A a ( k ) + u b A b ( k ) (cid:12)(cid:12) of the corresponding FB amplitudes A a,b ( k ) = dens( Λ a,b )vol( W a,b ) [ W a,b ( − k ⋆ )= dens( Λ )vol( W ) [ W a,b ( − k ⋆ ) = 1 √ [ W a,b ( − k ⋆ ) . The corresponding autocorrelation measure can oncemore be expressed in terms of pair correlation func-tions, now distinguishing points in Λ a and Λ b , ν αβ ( r ) := dens (cid:0) Λ α ∩ ( Λ β − r ) (cid:1) dens( Λ ) = ν βα ( − r ) . These coefficients are positive for all r ∈ Λ β − Λ α and vanish otherwise, and in particular satisfy therelation P α,β ∈{ a,b } ν αβ ( r ) = ν ( r ).The relation (9) between the FB coefficients andthe Fourier transform of the compact windows holdsfor any regular model set, which is a cut and projectset with some ‘niceness’ constraint on the window;see [16, Thm. 9.4] for details. While this works wellfor many of the nice examples with polygonal win-dows, it becomes practically impossible to computethe FB coefficients in this way if the windows arecompact sets with fractal boundaries. Such windowsnaturally arise for cut and project sets which also possess an inflation symmetry. Indeed, some of thestructure models of icosahedral quasicrystals, see [51]for an example, feature experimentally determinedwindows whose shapes may indicate first steps of afractal construction on the boundary.Let us therefore explain a different approach thatwill permit an efficient computation of the diffractionalso for such, more complicated, situations.5. Renormalisation and internal cocycle
Let us reconsider our motivating example, the Fi-bonacci point sets Λ a,b of Eq. (6). We will use boththeir inflation structure and their description as cutand project sets. Here, we make use of the itera-tion (5) and the corresponding relation (7) for thewindows (or, more precisely, the closure of the win-dows). This inflation structure induces the followingrelation between the characteristic functions of thewindows,(12) 1 W a = 1 σW a ∪ σW b and 1 W b = 1 σW a + σ , where we again set σ = τ ⋆ = 1 − τ . Since the (closed)windows only share at most boundary points, we ob-serve that 1 σW a ∪ σW b = 1 σW a + 1 σW b holds as anequality of L -functions. Now, we can apply theFourier transform, where it will turn out to be moreconvenient to work with the inverse Fourier trans-form from the start. Applying the transform yieldsthe relations(13) } W a = ~ σW a + ~ σW b and } W b = σW a + σ . Note that, by an elementary change of variable cal-culation, one has(14) αK + β ( y ) = | α | e π i βy | K ( αy )for arbitrary α, β ∈ R with α = 0 and any compactset K ⊂ R .Defining h a,b := ~ W a,b and using Eq. (14), we canrewrite Eq. (13) as(15) (cid:18) h a h b (cid:19) ( y ) = | σ | B ( y ) (cid:18) h a h b (cid:19) ( σy )with the matrix(16) B ( y ) := (cid:18) π i σy (cid:19) . The matrix B is obtained by first taking the ⋆ -map ofthe set-valued displacement matrix T of Eq. (4) andthen its inverse Fourier transform. For this reason, B is called the internal Fourier matrix [18], to distin-guish it from the Fourier matrix of the renormalisa-tion approach in physical space [9, 8]; see [25, 26] forvarious extensions with more flexibility in the choiceof the interval lengths.Going back to using Dirac notation, we introduce | h i = ( h a , h b ) T , which satisfies | h (0) i = τ | v i withthe right eigenvector | v i of the substitution matrix M from Eq. (2). Applying the iteration (15) n timesthen gives(17) | h ( y ) i = | σ | n B ( n ) ( y ) | h ( σ n y ) i where B ( n ) ( y ) := B ( y ) B ( σy ) · · · B ( σ n − y ) . In particular, these matrices satisfy B (1) = B and B ( n ) (0) = M n for all n ∈ N , where M is the substi-tution matrix from Eq. (1), as well as the relations(18) B ( n + m ) ( y ) = B ( n ) ( y ) B ( m ) ( σ n y )for any m, n ∈ N . Note that B ( n ) ( y ) defines a matrixcocycle, called the internal cocycle , which is relatedto the usual inflation cocycle (in physical space) byan application of the ⋆ -map to the displacement ma-trices of the powers of the inflation rule; compare[11, 18] and see [25, 26] for a similar approach. Notealso that | σ | <
1, which means that σ n approaches0 exponentially fast as n → ∞ . We can exploit thisexponential convergence to efficiently compute thediffraction amplitudes, which are essentially the ele-ments of the vector | h i .Considering the limit as n → ∞ in Eq. (17), onecan show that(19) | h ( y ) i = C ( y ) | h (0) i with(20) C ( y ) := lim n →∞ | σ | n B ( n ) ( y ) , which exists pointwise for every y ∈ R . In fact, onehas compact convergence, which implies that C ( y )is continuous [18, Thm. 4.6 and Cor. 4.7]. Clearly,since B ( n ) (0) = M n , we have C (0) = P with theprojector P = | v ih u | from Eq. (3).Using Eq. (18) with m = 1 and letting n → ∞ ,one obtains τ C ( y ) = C ( y ) M , since | σ | = τ − . This relation implies that each rowof C ( y ) is a multiple of the left eigenvector h u | of thesubstitution matrix M from Eq. (2), which meansthat we can define a vector-valued function | c ( y ) i such that(21) C ( y ) = | c ( y ) ih u | holds with | c ( y ) i = (cid:0) c a ( y ) , c b ( y ) (cid:1) T , where we have | c (0) i = | v i .From Eqs. (19) and (21), we obtain | h ( y ) i = | c ( y ) ih u | h (0) i = τ | c ( y ) i and thus the inverse Fourier transforms of the win-dows, ~ W a,b = h a,b , are encoded in the matrix C .For the Fibonacci case, we can calculate | c ( y ) i bytaking the Fourier transforms of the known windows W a,b to obtain c a ( y ) = e π i( τ − y − e π i( τ − y π i y and c b ( y ) = e π i( τ − y − e − π i y π i y . Note that these functions never vanish simultane-ously, so C ( y ) is always a matrix of rank 1. However,taking the Fourier transform of the windows takes usessentially back to the standard approach.The main benefit of the internal cocycle approachis that it applies in other situations, where no ex-plicit calculation of the (inverse) Fourier transformof the windows is feasible. This is achieved by ap-proximating C ( y ) by | σ | n B ( n ) ( y ) for a sufficientlylarge n , such that | σ | n y is small and C ( y ) is ap-proximated sufficiently well. This works because the(closed) windows are compact sets, so that their (in-verse) Fourier transforms are continuous functions.The convergence of this approximation is exponen-tially fast. We refer to [18] for further details and anextension of the cocycle approach to more generalinflation systems, and [19] for a planar example.From the general formula (9) for regular modelsets, the FB amplitudes are(22) A Λ a,b ( k ) = h a,b ( k ⋆ ) √ τ √ c a,b ( k ⋆ )for k ∈ L ⊛ . So, the relevant input is the knowledgeof the Fourier module, which determines where theBragg peaks are located. Then, one can approximate C by evaluating the matrix product in Eq. (20), forany chosen k ∈ L ⊛ , at y = k ⋆ and with a sufficientlylarge n . In what follows, numerical calculations andillustrations are based on this cocycle approach dueto its superior speed and accuracy in the presence ofcomplex windows.6. Fractally bounded windows
The internal cocycle approach of Section 5 wasfirst applied to a ternary inflation tiling with thesmallest Pisot–Vijayaraghavan (PV) number (alsoknown as the ‘plastic number’) as its inflation mul-tiplier [19]. In the cut and project description,the internal space of this one-dimensional tiling istwo-dimensional, and the windows are
Rauzy frac-tals [42]. This means that the windows are stilltopologically regular, so the closure of their interior,but have a fractal boundary of zero Lebesgue mea-sure. Consequently, the general diffraction result formodel sets still applies, and the diffraction is givenby the Fourier transform of the windows as describedabove. In turn, this means that the internal cocycleapproach applies and can be used to compute theFourier transforms and the diffraction intensities forsuch tilings; see [19] for details.Here, we discuss examples of planar projectiontilings with fractally bounded windows, which are
Figure 4.
Patch of the square Fibonacci tiling.based on direct product variations (DPVs) of Fi-bonacci systems, as recently described in [7]. Clearly,if one considers a direct product structure based onthe Fibonacci tiling, one obtains a tiling of the plane,called the square Fibonacci tiling . It is built fromfour prototiles, a large square of edge length τ , asmall square of edge length 1, and two rectangleswith a long ( τ ) and a short (1) edge; see Figure 4.As a direct product of an inflation tiling, this two-dimensional square Fibonacci tiling also possesses aninflation rule, which takes the form(23) where we labelled the small and large squares by 0and 3, and the two rectangles by 1 and 2, respec-tively. A DPV is now obtained by modifying theserules while keeping the stone inflation character in-tact. Clearly, there are two possibilities to rearrangethe images of the rectangles by swapping the two Figure 5.
Central part of the diffraction im-age of the square Fibonacci tiling. tiles, and a close inspection shows that there are alto-gether 12 ways of rearranging the image of the largesquare. This means that there are 48 distinct infla-tion rules in total, which all share these prototilesand the same inflation matrix.Due to the direct product structure, the squareFibonacci tiling clearly possesses a cut and projectdescription. The windows for the four prototilesare obtained as products of the original windows.The product structure thus extends to the diffractionmeasure, which is supported on the Fourier module L ⊛ × L ⊛ , where L ⊛ = √ Z [ τ ] is the Fourier module of theone-dimensional Fibonacci tiling. The diffractionamplitudes are also given by products of those for theone-dimensional system, and thus easy to compute.An illustration of the diffraction pattern is shownin Figure 5. Here, Bragg peaks are represented bydisks, centred at the position of the peak, with anarea proportional to its intensity.It turns out that all
48 DPV inflation tilings areregular model sets, and hence are pure point diffrac-tive; see [7, Thm. 5.2]. They all share the sameFourier module L ⊛ × L ⊛ . This implies that the Braggpeaks are always located at the same positions. How-ever, their intensities are determined by the Fouriertransform of the windows, and it turns out that thewindows of these DPVs can differ substantially.In particular, 20 of these DPVs possess windowsof Rauzy fractal type, of which there are three dif-ferent types, called ‘castle’, ‘cross’ and ‘island’ in [7].They have different fractal dimension of the windowboundaries, which are approximately 1 . . . for the large square. Note that this rule dissects theinflated large square such that there is a reflectionsymmetry along the main diagonal, which will be re-flected in a symmetry of the tiling (which maps the Figure 6.
Castle-type window for the DPV(24). The windows for the four types of tiles aredistinguished by colour, namely red (0), yellow(1), green (2) and blue (3). The outer boxesmark the square [ − τ, τ ] , with the coordinateaxes indicated as well.squares onto themselves and interchanges the rectan-gles). This is also apparent for the windows in Fig-ure 6. The windows for the large and small squaresare mapped onto themselves under reflection at themain diagonal, while the windows for the rectangulartiles are interchanged. The diffraction pattern alsorespects this symmetry; an illustration is shown inFigure 7.For the cross-type windows, the inflation of thelarge square is given by(25) which, in contrast to the previous example, has noreflection symmetry. Consequently, neither the win-dows shown in Figure 8 nor the diffraction imageillustrated in Figure 9 have any reflection symmetry. Figure 7.
Diffraction image of the DPV (24).
Figure 8.
Cross-type window for the DPV (25).The same is true for the final example with theisland-type window shown in Figure 10. This corre-sponds to the inflation(26) of the large square tile. The corresponding diffrac-tion pattern is illustrated in Figure 11.Comparing the diffraction patterns of Figures 7,9 and 11 with those of the square Fibonacci tilingshown in Figure 5, we note that the strongest peaksremain pretty much the same, while the intensitiesof the weaker peaks show some intriguing behaviour.For the fractally-bounded windows, one generallysees more peaks, which is due to the larger spreadof the window in internal space, and the slower as-ymptotic decay of the Fourier transform of the win-dow. With limited resolution, some of the intensitydistributions on these peaks could resemble contin-uous components, so might potentially be mistakenas such in experiments.
Figure 9.
Diffraction image of the DPV (25).
Figure 10.
Island-type window for the DPV (26).7.
Diffraction and hyperuniformity
The discovery of quasicrystals highlighted the lackof a clear definition of the concept of order . Incrystallography, diffraction is the main tool to de-tect long-range order, and a pure point diffraction isgenerally associated to an ordered, (quasi)crystallinestructure, while absolutely continuous diffraction istypically seen as an indication of random disorder(but see [29, 14, 27, 28] for examples of deterministicstructures that show absolutely continuous diffrac-tion). Here, we briefly discuss a related concept thathas recently gained popularity.From the original idea of using the degree of‘(hyper)uniformity’ in density fluctuations in many-particle systems [52] to characterise their order, the scaling behaviour of the total diffraction intensitynear the origin has emerged as a possible measureto capture long-distance correlations. As far as ape-riodic structures are concerned, there are in fact a
Figure 11.
Diffraction image of the DPV (26). number of early, partly heuristic, results in the lit-erature [35, 4, 31]. These have recently been refor-mulated and extended [40, 41] and rigorously estab-lished [17], using exact renormalisation relations forprimitive inflation rules [8, 9, 36, 11, 20, 37].For the investigation of scaling properties, we fol-low the existing literature and define(27) Z ( k ) := b γ (cid:0) (0 , k ] (cid:1) , which is a modified version of the distribution func-tion of the diffraction measure. Here, Z ( k ) is thetotal diffraction intensity in the half-open interval(0 , k ], and thus ignores the central peak. Due to thereflection symmetry of b γ with respect to the origin,this quantity can also be expressed as Z ( k ) = 12 (cid:16)b γ (cid:0) [ − k, k ] (cid:1) − b γ (cid:0) { } (cid:1)(cid:17) . The interest in the scaling of Z ( k ) as k → k behaviour ofthe diffraction measure probes the long-wavelengthfluctuations in the structure. As the latter is relatedto the variance in the distribution of patches, it canserve as an indicator for the degree of uniformity ofthe structure [52]. It is obvious that any periodicstructure leads to Z ( k ) = 0 for all sufficiently smallwave number k .Here, we review the result for variants of the one-dimensional Fibonacci model sets considered above,where we now allow for changes of the windows. Fora general discussion of this approach and more ex-amples of systems with different types of diffraction,we refer to [17] and references therein.Let us look at the diffraction for a cut and projectset with the same setup as the Fibonacci tiling con-sidered in Section 4, but with the window W replacedby an arbitrary finite interval of length s . Note thatthese tilings, in general, do not possess an inflationsymmetry. Nevertheless, the diffraction intensity isstill of the form (10), but now featuring the intervallength s , and is given by I ( k ) = I (0) (cid:0) sinc( πsk ⋆ ) (cid:1) for all k ∈ L ⊛ . Now, consider a sequence of posi-tions τ − ℓ k with k ∈ L ⊛ and ℓ ∈ N . Since we havesinc( x ) = sin( x ) /x = O ( x − ) as x → ∞ , it followsthat I ( τ − ℓ k ) = O (cid:0) τ − ℓ (cid:1) as ℓ → ∞ .Consequently, the sum of intensities along the se-ries of peaks, Σ ( k ) = ∞ X ℓ =0 I ( τ − ℓ k ) , satisfies the asymptotic behaviour Σ ( τ − ℓ k ) ∼ c ( k ) τ − ℓ Σ ( k ) as ℓ → ∞ , where it can be shown that c ( k ) = O (1)[17]. Expressing Z ( k ) in terms of these sums gives Z ( k ) = X κ ∈ L ⊛ kτ <κ k Σ ( k ) , which implies the asymptotic behaviour Z ( τ − ℓ k ) = τ − ℓ Z ( k ) . This leads to a scaling behaviour of the form Z ( k ) = O ( k ) as k ց s ∈ Z [ τ ]. This obviously holds for ouroriginal Fibonacci window W of length τ . However,one gets a stronger result for this case [17, 40], as weshall now recall.Choosing s ∈ Z [ τ ] means s = a + bτ with a, b ∈ Z .For 0 = k ∈ L ⊛ , set k = κ/ √ with κ = m + nτ forsome m, n ∈ Z , excluding m = n = 0. Applying the ⋆ -map then gives I ( τ − ℓ k ) = I (0) (cid:18) sinc (cid:16) πτ ℓ sκ ⋆ √ (cid:17)(cid:19) , with ℓ ∈ N .Now, denote by f n with n ∈ Z the Fibonacci num-bers defined by f = 0, f = 1 and the recursion f n +1 = f n + f n − . They satisfy the well-known for-mula(28) f n = 1 √ (cid:16) τ n − (cid:0) − /τ (cid:1) n (cid:17) for all n ∈ Z . Using this relation, we obtainsin (cid:16) πτ ℓ sκ ⋆ √ (cid:17) = sin (cid:16) π | sκ ⋆ | √ τ − ℓ (cid:17) = π ( sκ ⋆ ) τ − ℓ + O (cid:0) τ − ℓ (cid:1) (29)as ℓ → ∞ . Here, the first step follows by usingEq. (28) to replace τ ℓ / √ and then reducing the ar-gument via the relationsin( mπ + x ) = ( − m sin( x ) , which holds for all m ∈ Z and x ∈ R . This ispossible because all Fibonacci numbers are integers.The second step then uses the Taylor approximationsin( x ) = x + O ( x ) for small values of x .Now, the same argument as above implies the as-ymptotic behaviour Z ( τ − ℓ k ) ≍ τ − ℓ Z ( k ) , and hence Z ( k ) = O ( k ). This results means that,for inflation-invariant projection sets, the distribu-tion function Z ( k ) of the diffraction intensity van-ishes like k as k ց
0, while in the generic case wefind a k -behaviour. This example illustrates thatthe behaviour of the diffraction intensity near 0 can | W | = τ − − − − − − − − kI/I | W | = π/ − − − − − − − − kI/I | W | = √ − − − − − − − − kI/I Figure 12.
Double logarithmic plot of the in-tensity ratio
I/I of Bragg peaks located at k = ( m + nτ ) / √ | m | , | n | ) ,where I = I (0), for windows W of differentlengths. The dashed line corresponds to k for | W | = τ (top) and to k for the other two cases.pick up non-trivial aspects of order in this system.This is illustrated for some cases in Figure 12.Let us briefly comment on the scaling behaviourfor other prominent examples of aperiodic order dis-cussed in [17]. For noble means inflations, we ob-serve the same k -scaling as for the Fibonacci tiling.The period doubling sequence, which is limit peri-odic, shows k -scaling, and a range of scaling expo-nents is accessible for substitutions of more than twoletters. For the Thue–Morse sequence, which is theparadigm of an inflation structure with singular con-tinuous diffraction, we do not obtain a power law,but an exponential scaling behaviour which decaysfaster than any power; see also [12] for more on thescaling of the spectrum for this system. Finally, theRudin–Shapiro sequence, which has absolutely con-tinuous spectrum, shows a linear scaling behaviour,due to the constant density of its diffraction measure. Acknowledgements
It is our pleasure to thank Claudia Alfes-Neumann, Natalie Priebe Frank, Neil Ma˜nibo, BerndSing, Nicolae Strungaru and Venta Terauds for valu-able discussions. This work was supported by theGerman Research Foundation (DFG), within theCRC 1283 at Bielefeld University, and by EPSRC,through grant EP/S010335/1.
References [1] Akiyama, S. & Arnoux, P. (eds.) (2020).
Tiling Dy-namical Systems: Substitutions and Beyond . Berlin:Springer, in press.[2] Akiyama, S., Barge, M., Berth´e, V., Lee, J.-Y., Siegel,A. (2015). On the Pisot substitution conjecture. In: Kel-lendonk, J., Lenz, D., Savinien, J. (eds.)
Mathematicsof Aperiodic Order . Basel: Birkh¨auser, pp 33–72.[3] Allouche, J.-P. & Shallit, J. (2003).
Automatic Se-quences . Cambridge: Cambridge University Press.[4] Aubry, S., Godr`eche, C. & Luck, J.M. (1988). Scal-ing properties of a structure intermediate betweenquasiperiodic and random,
J. Stat. Phys. , 1033–1074.[5] Baake, M., Coons, M. & Ma˜nibo, N. (2020). Binaryconstant-length substitutions and Mahler measures ofBorwein polynomials. In: Bailey, D., Borwein, N.S.,Brent, R.P., Burachik, R.S., Osborn, J.-A.H., Sims, B.,Zhu, Q.J. (eds.) From Analysis to Visualization: JBCC2017 . Cham: Springer, pp 303–322.[6] Baake, M., Ecija, D. & Grimm, U. (2016). A guide tolifting aperiodic structures.
Z. Kristallogr. , 507-515.[7] Baake, M., Frank, N.P. & Grimm, U. (2021). Three vari-ations on a theme by Fibonacci.
Stoch. Dyn. , in press.arXiv:1910.00988.[8] Baake, M., Frank, N.P., Grimm, U. & Robinson, E.A.(2019). Geometric properties of a binary non-Pisot in-flation and absence of absolutely continuous diffraction,
Studia Math. , 109–154.[9] Baake, M. & G¨ahler, F. (2016). Pair correlations of ape-riodic inflation rules via renormalisation: Some interest-ing examples.
Topology & Appl. , 4–27.[10] Baake, M., G¨ahler, F. & Grimm, U. (2013). Examplesof substitution systems and their factors
J. Integer Seq. , 13.2.14:1–18.[11] Baake, M., G¨ahler, F. & Ma˜nibo, N. (2019). Renormali-sation of pair correlation measures for primitive inflationrules and absence of absolutely continuous diffraction. Commun. Math. Phys. , 591–635.[12] Baake, M., Gohlke, P., Kesseb¨ohmer, M. & Schindler, T.(2019). Scaling properties of the Thue–Morse measure.
Discr. Cont. Dynam. Syst. A , 4157–4185.[13] Baake, M. & Grimm, U. (2003). A note on shelling. Discr. Comput. Geom. , 573–589.[14] Baake, M. & Grimm, U. (2009). Kinematic diffractionis insufficient to distinguish order from disorder. Phys.Rev. B , 020203:1–4 and (2009) 029903(E) (Erra-tum).[15] Baake, M. & Grimm, U. (2012). Mathematical diffrac-tion of aperiodic structures. Chem. Soc. Rev. , 6821–6843.[16] Baake, M. & Grimm, U. (2013). Aperiodic Order. Vol-ume 1: A Mathematical Invitation . Cambridge: Cam-bridge University Press. [17] Baake, M. & Grimm, U. (2019). Scaling of diffraction in-tensities near the origin: Some rigorous results.
J. Stat.Mech.: Theory Exp. , 054003:1–25.[18] Baake, M. & Grimm, U. (2019). Fourier transform ofRauzy fractals and point spectrum of 1D Pisot inflationtilings.
Preprint arXiv:1907.11012.[19] Baake, M. & Grimm, U. (2020). Diffraction of a modelset with complex windows.
J. Phys.: Conf. Ser.
Lett. Math.Phys. , 1783–1805.[21] Baake, M., Huck, C. & Strungaru, N. (2017). On weakmodel set of extremal density.
Indag. Math. , 3–31.[22] Baake, M. & Zeiner, P. (2017). Geometric enumera-tion problems for lattices and embedded Z -modules.In: Baake, M., Grimm U. (eds.) Aperiodic Order.Vol. 2: Crystallography and Almost Periodicity . Cam-bridge: Cambridge University Press, pp 73–172.[23] Bartlett, A. (2018). Spectral theory of Z d substitutions. Ergodic Th. & Dynam. Syst. , 1289–1341.[24] Berlinkov, A. & Solomyak, B. (2019). Singular substitu-tions of constant length. Ergodic Th. & Dynam. Syst. , 2384–2402.[25] Bufetov, A. & Solomyak, B. (2018). A spectral cocy-cle for substitution systems and translation flows. J.d’Analyse , in press. arXiv:1802.04783.[26] Bufetov, A. & Solomyak, B. (2020). On singular substi-tution Z -actions. Preprint arXiv:2003.11287.[27] Chan, L. & Grimm, U. (2017). Spectrum of a Rudin–Shapiro-like sequence.
Adv. Appl. Math. , 16–23.[28] Chan, L., Grimm, U. & Short, I. (2018). Substitution-based structures with absolutely continuous spectrum. Indag. Math. , 1072–1086.[29] Frank, N.P. (2003). Substitution sequences in Z d with anon-simple Lebesgue component in the spectrum, Ergod.Th. & Dynam. Syst. , 519–532.[30] Frettl¨oh, D. (2017). More inflation tilings. In: Baake, M.and Grimm, U. (eds.) Aperiodic Order. Vol. 2: Crys-tallography and Almost Periodicity . Cambridge: Cam-bridge University Press, pp 1–37.[31] Godr`eche, C. & Luck, J.M. (1990). Multifractal analysisin reciprocal space and the nature of the Fourier trans-form of self-similar structures,
J. Phys. A: Math. Gen. , 3769–3797.[32] Grimm, U. (2015). Aperiodic crystals and beyond. ActaCrystallogr. B , 258–274.[33] Hof, A. (1995). On diffraction by aperiodic structures. Commun. Math. Phys. , 25–43.[34] Kellendonk, J., Lenz, D. & Savinien, J. (eds.) (2015).
Mathematics of Aperiodic Order . Basel: Birkh¨auser.[35] Luck, J.M. (1993). A classification of critical phenomenaon quasi-crystals and other aperiodic structures,
Euro-phys. Lett. , 359–364.[36] Ma˜nibo, N. (2017). Lyapunov exponents for binarysubstitutions of constant length. J. Math. Phys. ,113504:1–9.[37] Ma˜nibo, C.N. (2019). Lyapunov Exponents in theSpectral Theory of Primitive Inflation Systems , PhDthesis, Bielefeld University (2019); available at urn:nbn:de:0070-pub-29359727 .[38] Moody, R.V. (2000). Model sets: A survey. In: Axel, F.,D´enoyer, F. and Gazeau, J.P. (eds.)
From Quasicrystalsto More Complex Systems , Berlin: Springer, pp 145–166. [39] Moody, R.V. (2002). Uniform distribution in model sets, Can. Math. Bull. , 123–130.[40] O˘guz, E.C., Socolar, J.E.S., Steinhardt, P.J. &Torquato, S. (2017). Hyperuniformity of quasicrystals, Phys. Rev. B , 054119:1–10.[41] O˘guz, E.C., Socolar, J.E.S., Steinhardt, P.J. &Torquato, S. (2019). Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings, Acta Cryst. A , 3–13.[42] Pytheas Fogg, N. (2002). Substitutions in Dynamics,Arithmetics and Combinatorics , LNM 1794. Berlin:Springer.[43] Queff´elec M. (2010).
Substitution Dynamical Systems —Spectral Analysis , LNM 1294, 2nd ed. Berlin: Springer.[44] Richard, C. & Strungaru, N. (2017). A short guide topure point diffraction in cut-and-project sets.
J. Phys.A: Math. Theor. , 154003:1–25.[45] Richard, C. & Strungaru, N. (2017). Pure point diffrac-tion and Poisson summation. Ann. Henri Poincar´e ,3903–3931.[46] Schlottmann, M. (2000). Generalised model sets and dy-namical systems. In: Baake, M. and Moody, R.V. (eds.) Directions in Mathematical Quasicrystals . Providence:Amer. Math. Soc., pp 143–159.[47] Shechtman, D., Blech, I., Gratias, D. & Cahn, J.W.(1984). Metallic phase with long-range orientational or-der and no translational symmetry.
Phys. Rev. Lett. ,1951–1953.[48] Solomyak, B. (1997). Dynamics of self-similar tilings, Ergod. Th. & Dynam. Syst. , 695–738 and Ergod. Th.& Dynam. Syst. (1999) and 1685 (Erratum).[49] Strungaru, N. (2017). Almost periodic pure point mea-sures. In: Baake, M., Grimm U. (eds.) Aperiodic Order.Vol. 2: Crystallography and Almost Periodicity . Cam-bridge: Cambridge University Press, pp 271–342.[50] Strungaru, N. (2020). On the Fourier analysis of mea-sures with Meyer set support.
J. Funct. Anal. ,108404:1–30.[51] Takakura, H., G´omez, C.P., Yamamoto, A., de Boissieu,M. & Tsai, A.P. (2007). Atomic structure of the binaryicosahedral Yb–Cd quasicrystal.
Nature Mat. , 53–63.[52] Torquato, S. & Stillinger, F.H. (2003). Local densityfluctuations, hyperuniformity, and order metrics, Phys.Rev. E68