Diffraction Theory of Tiling Dynamical Systems Michael Baake Faculty of Mathematics, Bielefeld ( joint work with Uwe Grimm) Workshop on Fractals and Tilings, Strobl, July 2009 p.1
Menue Diffraction theory Thue-Morse chain Autocorrelation Diffraction Fourier series Volterra iteration TM measure Gen. Morse systems 1 3/4 1/2 1/4 F(x) 0 1/3 2/3 1 x Workshop on Fractals and Tilings, Strobl, July 2009 p.2
Diffraction theory Structure: translation bounded measure ω assumed amenable Autocorrelation: γ = γ ω = ω ω := lim R ω R ω R vol(b R ) Diffraction: γ = γ pp + γ sc + γ ac (relative to λ) pp: Bragg peaks only (lattices, model sets) ac: diffuse scattering with density sc: whatever remains... Workshop on Fractals and Tilings, Strobl, July 2009 p.3
Thue-Morse chain Substitution: : 1 1 1 1 11 ( 1 ˆ= 1 ) Iteration and fixed point: 1 1 1 1 1 11 1 1 11 111 1... v = (v) = v 0 v 1 v 2 v 3... v 2i = v i and v 2i+1 = v i v is (strongly) cube-free hull of v is aperiodic and strictly ergodic sum of the binary digits of v i = ( 1) i Two-sided version: w i = { v i, for i 0 v i 1, for i < 0 Workshop on Fractals and Tilings, Strobl, July 2009 p.4
Autocorrelation Autocorrelation: γ = lim n 1 2n+1 with ω n = ( ω n ω n ) n i= n w i δ i Structure: γ = m Z η(m)δ m with η(m) = lim n 1 n n 1 v i v i+m i=0 and η( m) = η(m) for m 0 Recursion: η(0) = 1, η(1) = 3 1 and, for all m 0, η(2m) = η(m) and η(2m + 1) = 1 ( ) 2 η(m) + η(m + 1) Workshop on Fractals and Tilings, Strobl, July 2009 p.5
Diffraction: Absence of pp part γ = µ δ Z with µ = γ [0,1) and η(m) = 1 0 e2πimy dµ(y) (Herglotz-Bochner) Wiener s criterion: µ pp = 0 Σ(N) = o(n) where Σ(N) = N ( ) 2 m= N η(m) Argument: Σ(4N) 3 2Σ(2N) (by recursion for η) = µ = µ cont = µ sc + µ ac Define: F(x) = µ ( [0,x] ) for x [0, 1], where F = F ac + F sc Workshop on Fractals and Tilings, Strobl, July 2009 p.6
Diffraction: Absence of ac part Functional relation: df ( ) ( x 2 +df x+1 ) 2 = df(x) df ( ) ( x 2 df x+1 ) 2 = cos(πx) df(x) valid for F ac and F sc separately (µ ac µ sc ) Define: η ac (m) = 1 0 e2πimx df ac (x) same recursion as for η(m), but η ac (0) free Riemann-Lebesgue lemma: lim m ± η ac (m) = 0 = η ac (0) = 0 = η ac (m) 0 = F ac = 0 Theorem: µ = µ sc and γ is purely sc. (Fourier uniqueness thm) Workshop on Fractals and Tilings, Strobl, July 2009 p.7
Fourier series Functional equation: F(1 x) + F(x) = 1 on [0, 1] and F(x) = 1 2 2x 0 ( 1 cos(πy) ) df(y) for x [0, 1 2 ] = F(x) = x + m 1 η(m) mπ sin(2πmx) F(x) x continuous and of bounded variation Uniformly convergent Fourier series Unique solution (contraction argument) F strictly increasing = supp(µ) = [0, 1] Workshop on Fractals and Tilings, Strobl, July 2009 p.8
Volterra iteration Define: F 0 (x) = x and F n+1 (x) = 1 2 2x 0 ( 1 cos(πy) ) F n (y) d(y) for n 0 and x [0, 1 2 ], extension to [0, 1] by symmetry = df n (x) = g n (x) dx = g n (x) = n 1 ( 1 cos(2 k+1 πx) ) k=0 Riesz product: sequence of ac measures, vague convergence to µ = µ sc Workshop on Fractals and Tilings, Strobl, July 2009 p.9
TM measure 1 0.5 0 0 0.5 1 Workshop on Fractals and Tilings, Strobl, July 2009 p.10
Generalised Morse sequences Substitution: : 1 1 k 1 l 1 1 k 1 l (with k,l N) Fixed point: v 0 = 1, v m(k+l)+r = { vm, if 0 r < k v m, if k r < k + l Coefficients: η ( (k + l)m + r ) = 1 k+l η(0) = 1, η(1) = k+l 3 k+l+1, and with m N 0, 0 r k + l 1, and ( αk,l,r η(m) + α k,l,k+l r η(m + 1) ) α k,l,r = k + l r 2 min(k,l,r,k + l r) Workshop on Fractals and Tilings, Strobl, July 2009 p.11
Generalised Morse sequences, ctd Fourier series: F(x) = γ ( [0,x] ) = x + m 1 η(m) m π sin(2πmx) (uniform convergence) Riesz product: ϑ ( (k + l) n x ) n 0 ϑ(x) = 1 + 2 k + l with k+l 1 r=1 α k,l,r cos(2πrx) (vague convergence) Workshop on Fractals and Tilings, Strobl, July 2009 p.12
Period doubling sequences Block map: ψ : 1 1, 11 a, 11, 1 1 b gen. period doubling: : X TM ψ X pd π Sol 2 (k+l) X TM ψ (2:1) X pd π (a.e. 1:1) Sol a b k 1 ab l 1 b b b k 1 ab l 1 a coincidence = model set (Dekking) Workshop on Fractals and Tilings, Strobl, July 2009 p.13
References N. Wiener, The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions. Part I: The spectrum of an array, J. Math. Massachusetts 6 (1927) 145 157 K. Mahler, dto. Part II: On the translation properties of a simple class of arithmetical functions, J. Math. Massachusetts 6 (1927) 158 163 M. Keane, Generalized Morse sequences, Z. Wahrscheinlichkeitsth. Verw. Gebiete 10 (1968) 335 353 S. Kakutani, Strictly ergodic symbolic dynamical systems, in: Proc. 6th Berkeley Symposium on Math. Statistics and Probability, eds. L.M. LeCam, J. Neyman and E.L. Scott (eds.), Univ. of California Press, Berkeley (1972), pp. 319 326 F.M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitsth. verw. Geb. 41 (1978) 221 239 M. Queffélec, Substitution Dynamical Systems Spectral Analysis, LNM 1294, Springer, Berlin (1987) M. Baake and U. Grimm, The singular continuous diffraction measure of the Thue-Morse chain, J. Phys. A.: Math. Theor. 41 (2008) 422001; arxiv:0809.0580 (math.ds) Workshop on Fractals and Tilings, Strobl, July 2009 p.14