The paradigm for singular continuous spectra in symbolic dynamics and in
mathematical diffraction is provided by the Thue-Morse chain, in its
realisation as a binary sequence with values in $\{\pm 1\}$. We revisit this
example and derive a functional equation together with an explicit form of the
corresponding singular continuous diffraction measure, which is related to the
known representation as a Riesz product.Comment: 6 pages, 1 figure; revised and improved versio

Allouche J-P

Bellissard J

Gazeau J-P Verger-Gaugry J-L

Kakutani S Le Cam L M Neyman J Scott E L

Katznelson Y

Lang S

Michael Baake

Queffélec M

Reed M

Uwe Grimm

English

arXiv

Baake M, Grimm U. The singular continuous diffraction measure of the Thue-Morse chain. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 2008;41(42).The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realization as a binary sequence with values in {+/- 1}. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product

Baake, Michael

Grimm, Uwe

Publications at Bielefeld University

The singular continuous diffraction measure of the Thue-Morse chain

The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realization as a binary sequence with values in ±1. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product

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Open Research OnlineThe Open University’s repository of research publicationsand other research outputsThe singular continuous diffraction measure of theThue-Morse chainJournal ItemHow to cite:Baake, Michael and Grimm, Uwe (2008). The singular continuous diffraction measure of the Thue-Morsechain. Journal of Physics A: Mathematical and Theoretical, 41(42) p. 422001.For guidance on citations see FAQs.c© 2008 Institute of PhysicsVersion: Accepted ManuscriptLink(s) to article on publisher’s website:http://dx.doi.org/doi:10.1088/1751-8113/41/42/422001Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyrightowners. For more information on Open Research Online’s data policy on reuse of materials please consult the policiespage.oro.open.ac.ukTHE SINGULAR CONTINUOUS DIFFRACTION MEASUREOF THE THUE-MORSE CHAINMICHAEL BAAKE AND UWE GRIMMAbstract. The paradigm for singular continuous spectra in symbolic dynamics and inmathematical diffraction is provided by the Thue-Morse chain, in its realisation as a bi-nary sequence with values in {±1}. We revisit this example and derive a functional equationtogether with an explicit form of the corresponding singular continuous diffraction measure,which is related to the known representation as a Riesz product.The Thue-Morse chain can be defined via the primitive substitution rule% :1 7→ 11¯1¯ 7→ 1¯1on the two-letter alphabet {1, 1¯}; see [1, 10] for background. Let v = v0v1v2 . . . be the uniqueone-sided fixed point of % with v0 = 1. This infinite word satisfies(1) v2n = vn and v2n+1 = v¯nfor all n ∈ N0, where 1¯ = 1. Consider the bi-infinite word w, defined aswn ={vn, for n ≥ 0,v−n−1, for n < 0,which is the unique reflection-symmetric fixed point of %2 with (admissible) seed w−1w0 = 11.It is cube-free and thus aperiodic. If S denotes the (two-sided) shift, defined by (Sw)n =wn+1, the Thue-Morse hull is the compact space X = {Smw | m ∈ Z}, where the closureis taken in the obvious product topology. Note that X coincides with the (discrete) localindistinguishability (or LI) class of w, and that the topological dynamical system (X, S) isstrictly ergodic; see [10] for details.Here, we consider the weighted Dirac comb(2) ω =∑n∈Zwn δn,where δx denotes the normalised Dirac measure on R, located at x, and 1¯ = −1. In particular,ω is a translation bounded, signed measure on R. The corresponding autocorrelation (orPatterson) measure γ, obtained by a volume-averaged convolution [6, 2] of ω with its reflectedcounterpart, reads(3) γ = η δZ :=∑m∈Zη(m) δm ,2 MICHAEL BAAKE AND UWE GRIMMwith the coefficients η(m) obtained as the limitsη(m) = limN→∞12N + 1∑−N≤k,`≤Nk−`=mwk w` = limN→∞12N + 1N∑n=−Nwn wn−m .Note that these limits exist due to unique ergodicity, compare [10], with η(0) = 1. Theautocorrelation coefficients satisfy η(−m) = η(m) and, due to (1), the recursions(4) η(2m) = η(m) and η(2m + 1) = −12(η(m) + η(m + 1)),valid for all m ≥ 0. In particular, η(1) = − 13 , and all coefficients are uniquely specified.By construction, η is a positive definite function on Z, wherefore the Herglotz-Bochnertheorem [8, Thm. I.7.6] guarantees the existence of a finite positive measure ν on [0, 1) with(5) η(m) =∫ 10e2piimy dν(y).The diffraction measure γ̂ of the weighted Thue-Morse comb ω of (2) is the Fourier transformof γ. An elementary calculation shows thatγ̂ = ν ∗ δZ and ν = γ̂∣∣[0,1).In this formulation, the spectral properties of γ̂ follow immediately from those of ν. Inparticular, ν has a unique decompositionν = νpp + νsc + νacinto its pure point, singular continuous and absolutely continuous parts (relative to Lebesguemeasure λ); see [12, Thms. I.13 and I.14].It was first shown by Kakutani [7] that ν is a purely singular continuous measure, soν = νsc. Let us briefly adapt this to our setting. By Wiener’s criterion [8, Cor. 7.11], νpp = 0is equivalent to limN→∞12N+1Σ(N) = 0, whereΣ(N) :=N∑m=−N(η(m))2.When N ≥ 1, we can use the recursion relations (4) to derive Σ(4N) ≤ 32 Σ(2N). Withα = log2(3/2) < 1, one then obtains the estimate1NΣ(N) = O(1/N1−α). At this point, weknow that ν = νsc + νac, and define the distribution function(6) F (x) := ν([0, x]),which is a continuous function of bounded variation on [0, 1]. It satisfies F (0) = 0 andF (x) + F (1− x) = 1, the latter as a consequence of the symmetry of ν.Following [7] and viewing ν as a Lebesgue-Stieltjes measure [9, Ch. X] with distributionfunction F , the two recursion relations (4) imply the identitiesdF (x2 ) + dF (x+12 ) = dF (x),dF (x2 )− dF (x+12 ) = − cos(pix) dF (x),(7)DIFFRACTION OF THE THUE-MORSE CHAIN 3for all x ∈ [0, 1]. The left-hand sides are obtained from the corresponding sides of therecursions by a change of variables, followed by a split of the new integration region intotwo parts. The actual equality of the measures follows because we obtain equality of theintegrals over arbitrary trigonometric polynomials, whence the Fourier uniqueness theorem[8, Thm. I.2.7] applies. The relations (7) must also hold for the two continuous componentsof ν separately, because νsc ⊥ νac by [9, Thm. VII.2.4].Writing νac = gλ with g ∈ L1([0, 1]), the identities (7) result in12(g(x2 ) + g(x+12 ))= g(x) and 12(g(x2 )− g(x+12 ))= − cos(pix) g(x),this time for almost all x ∈ [0, 1]. Defining now ηac(m) =∫ 10 e2piimxg(x) dx, we haveηac(−m) = ηac(m) and inherit a set of recursions identical to (4), however with ηac(0) = 0 asa result of the Riemann-Lebesgue lemma [8, Thm. I.2.8], and hence ηac(m) = 0 for all m ∈ Z.By the Fourier uniqueness theorem [8, Thm. I.2.7], this implies g(x) = 0 almost everywhere,and hence νac = 0. Since ν itself cannot be the zero measure, we have ν = νsc 6= 0, andthe Thue-Morse diffraction measure γ̂ is a purely singular continuous, Z-periodic, positivemeasure. It is the same for all Dirac combs of members of the hull X.The remainder of this note is concerned with an explicit calculation of ν, via its distributionfunction F , which does not seem to have been calculated in the literature so far. Adding thetwo equations for dF from (7), followed by integration, results in the functional equation(8) F (x) =12∫ 2x0(1− cos(piy))dF (y)for the continuous function F , valid for x ∈ [0, 12 ]. Since F is a continuous non-decreasingfunction on [0, 1], the difference f(x) = F (x)− x defines a continuous function f of boundedvariation, with f(0) = 0 and f(x) + f(1 − x) = 0 for all x ∈ [0, 1]. As such, f possesses auniformly converging Fourier series of the formf(x) =∑m≥1bm sin(2pimx),with bm = 2∫ 10 sin(2pimx) f(x) dx. From Eqs. (5) and (6), one finds η(m) = pim bm, so that(9) F (x) = x +∑m≥1η(m)mpisin(2pimx)is a uniformly converging Fourier series representation of the Thue-Morse distribution func-tion. This function satisfies the symmetry relation F (x) + F (1− x) = 1 for all x ∈ [0, 1], andis a solution of the integral equation (8).To interpret (8) in a wider setting, let us introduce the space(10) D ={G ∈ C([0, 1], R) ∣∣∣ G(0) = 0, G non-decreasing andG(x) + G(1− x) = 1 on [0, 1]}of continuous, non-decreasing distribution functions on [0, 1] with the required symmetry, anddefine the mapping Φ : D −→ D via G 7→ Φ(G) with(Φ(G))(x) ={12∫ 2x0(1− cos(piy))dG(y), if 0 ≤ x ≤ 12 ,1−(Φ(G))(1−x), if 12 < x ≤ 1.4 MICHAEL BAAKE AND UWE GRIMMOne can show that F of (9) is the only solution of Φ(F ) = F within D, because Φ is a weakcontraction with respect to the metric d(G, H) := V (G − H), where V denotes the totalvariation of continuous functions on [0, 1]; see [9, Ch. X] for background.For a numerical high-precision computation of F , we employ a Volterra-type iterationwithin D, which is superior to using (9) directly. Starting from F0(x) = x, where F0 ∈ D, weset Fn+1 = Φ(Fn) for n ≥ 0. This defines the (uniformly converging) sequence of distributionfunctions in D given by(11) Fn(x) = x +2n−1∑m=1c(n)mmpisin(2pimx).Here, the coefficients c(n)m satisfy the recursionsc(n+1)2m = c(n)m and c(n+1)2m+1 = −12(c(n)m + c(n)m+1)for n ≥ 1 and 0 ≤ m ≤ 2n − 1, together with c(1)0 = 0, c(1)1 = −12 and c(n+1)1 = −12(1 + c(n)1).They approach the autocorrelation coefficients η(m) via limn→∞ c(n)m = η(m) for arbitrarym ∈ N.By construction, the distribution functions Fn represent absolutely continuous measures(although their limit does not). Writing dFn(x) = fn(x) dx, one finds f0(x) = 1, and thefunctional equation (8) induces the recursionfn+1(x) =(1− cos(2pix))fn(2x) = 2(sin(pix))2fn(2x)for n ≥ 0. This gives the well-known explicit representation(12) fn(x) =n−1∏`=0(1− cos(2`+1pix))= 2nn−1∏`=0(sin(2`pix))2of the Thue-Morse measure as a Riesz product; compare [11, Sec. 1.4.2].The Volterra iteration leads to a sequence (Fn)n∈N0of continuous distribution functionsthat converge uniformly (on [0, 1]) to F , which is continuous as well. The latter is shown inFigure 1 and resembles the classic middle-thirds Cantor measure in various aspect, though ithas full support (see below). Note that the corresponding sequence (dFn)n∈N0of absolutelycontinuous measures is only vaguely convergent, with the limit being purely singular continu-ous. Therefore, it is somewhat misleading to show a density for the Thue-Morse measure, asis often found in the literature. Still, it may be instructive to inspect the sequence of densitiesfn in order to get some intuition on the singular nature of the Thue-Morse measure, or tostudy some of its scaling properties; see [5] and references therein.On initial inspection, the distribution function F seems to have a plateau around x = 12 ,with exact value 12 . More generally, as suggested by the Riesz products (12), one might expecta plateau around any x ∈({0, 1}∪{m/2k | k ∈ N, m ∈ N odd})∩ [0, 1], because the densitiesfn vanish at x for all sufficiently large n, with the order of this zero linearly increasing with n.However, these potential gaps are all closed (see below). The corresponding values of F canbe calculated with the series expansion (9). Unlike the situation of the gap labelling theoremfor one-dimensional Schro¨dinger operators [3], where one knows the values of the integratedDIFFRACTION OF THE THUE-MORSE CHAIN 50 0.5 100.51Figure 1. Distribution function F of the purely singular continuous diffrac-tion measure of the Thue-Morse chain in the interval [0, 1]. Note that F is astrictly increasing function, despite the appearance of plateau-like regions.density of states (IDOS) on the (always non-overlapping) plateaux but not their positions,we know the possible locations of the plateaux, but cannot see a topological constraint forthe corresponding values of the distribution function (which can be determined from (9)).It is interesting to note that the set of potential plateau locations coincides with the set ofpotential (but in our case extinct) Bragg peak positions, so the (extinct) Bragg peaks appearto ‘repel’ the continuous diffraction spectrum. Nevertheless, one hassupp( dF ) = supp(ν) = [0, 1]by [4, Prop. 28], which also implies that F is a strictly increasing function. In particular,as one can immediately see from (8), F (x) = 0 forces F (2x) = 0, which (when repeated)contradicts F (1/2) = 1/2 unless x = 0. This shows that there is no gap around 0 (and nonearound 1 by symmetry). The general argument uses that ν is a regular Borel measure; see[4] and references therein for details.6 MICHAEL BAAKE AND UWE GRIMMThe methods used above can also be applied to other substitutions of constant length thatfail to have pure point spectrum. This can be decided on the basis of Dekking’s criterion;see [11, Sec. 6] for details. Although one still has to check Wiener’s criterion and to find theanalogue of the functional equation (7), this approach seems worth pursuing.Acknowledgements. It is a pleasure to thank J. Bellissard, N.P. Frank, R.V. Moody andB. Solomyak for discussions, and the School of Mathematics and Physics at the Universityof Tasmania for their kind hospitality. This work was supported by the German ResearchCouncil (DFG), within the CRC 701, and by EPSRC via Grant EP/D058465.References[1] Allouche J.-P. and Shallit J. (1999). The ubiquitous Prouhet-Thue-Morse sequence. In Sequences and theirapplications, eds. Ding C., Helleseth T. and Niederreiter H., (Springer, London), pp. 1–16.[2] Baake M. and Moody R.V. (2004). Weighted Dirac combs with pure point diffraction, J. reine angew.Math. (Crelle) 573, 61–94; arXiv:math.MG/0203030.[3] Bellissard J., Herrmann D.J.L. and Zarrouati M. (2000). Hulls of aperiodic solids and gap labeling the-orems, in Directions in Mathematical Quasicrystals, eds. Baake M. and Moody R.V., CRM MonographSeries, vol. 13 (AMS, Rhode Island), pp. 207–258.[4] Benedetto J.J., Bernstein E. and Konstantinidis I. (2005). Multiscale Riesz products and their supportproperties, Acta Appl. Math. 88, 201–227.[5] Gazeau J.-P. and Verger-Gaugry J.-L. (2008). On the spectrum of the Thue-Morse quasicrystal and therarefaction phenomenon, preprint 710 (Institut Fourier, Paris).[6] Hof A. (1995). On diffraction by aperiodic structures, Commun. Math. Phys. 169, 25–43.[7] Kakutani S. (1972). Strictly ergodic symbolic dynamical systems. in Proceedings of the Sixth BerkeleySymposium on Mathematical Statistics and Probability, eds. Le Cam L.M., Neyman J. and Scott E.L.,(University of California Press, Berkeley), pp. 319–326.[8] Katznelson Y. (2004). An Introduction to Harmonic Analysis, 3rd ed. (Cambridge University Press, Cam-bridge).[9] Lang S. (1993). Real and Functional Analysis, 3rd ed. (Springer, New York).[10] Queffe´lec M. (1987). Substitution Dynamical Systems – Spectral Analysis, LNM 1294 (Springer, Berlin).[11] Queffe´lec M. (1995). Spectral study of automatic and substitutive sequences, in Beyond Quasicrystals,eds. Axel F. and Gratias, D. (Springer, Berlin, and EDP Sciences, Les Ulis), pp. 369–414.[12] Reed M. and Simon B. (1980). Methods of Modern Mathematical Physics. I: Functional Analysis, 2nd ed.(Academic Press, New York).Fakulta¨t fu¨r Mathematik, Universita¨t Bielefeld,Postfach 100131, 33501 Bielefeld, GermanyE-mail address: mbaake@math.uni-bielefeld.deDepartment of Mathematics and Statistics, The Open University,Walton Hall, Milton Keynes MK7 6AA, United KingdomE-mail address: u.g.grimm@open.ac.uk

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The singular continuous diffraction measure of the Thue–Morse chain

The singular continuous diffraction measure of the Thue-Morse chain

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