Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe...
Transcript of Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe...
Spectral Properties ofAperiodic Structures
Uwe Grimm
School of Mathematics & StatisticsThe Open UniversityMilton Keynes, UK
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.1
Perspective
Harmonic Analysis Dynamical Systems
ց ւ
Alg. / Comb. → Aperiodic Order ← Topology
ր տ
Number Theory Discrete Geometry
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Origins in crystallography
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Origins in crystallography
(Photo courtesy of the Ames Laboratory)
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Origins in crystallography5-fold
3-fold
2-fold
from: D. Shechtman, I. Blech, D. Gratias and J.W. Cahn (1984), Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53, 1951–1953
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Origins in crystallography
Wolf Prize in Physics 1999Nobel Prize in Chemistry 2011
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Diffraction theory
L A S E R
Wiener’s diagram obstacle f(x), with f(x) := f(−x)
f∗−−−→ f ∗ f
Fy
yF
f| . |2−−−→ |f |2
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Diffraction theory
Structure translation bounded measure ω
assumed ‘self-amenable’ (Hof 1995)
Autocorrelation γ = γω = ω ⊛ ω := limR→∞
ω|R ∗ ω|Rvol(BR)
Diffraction γ =(γ)pp
+(γ)sc
+(γ)ac
(relative to λ)
pp: Bragg peaks
ac: diffuse scattering with density
sc: whatever remains ...
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.4
Diffraction theory
Setting ω y γ = ω ⊛ ω xy γ 6y ω
Dirac comb on Z (similarly for lattices)
ω =∑
n∈Z
w(n) δn y γ =∑
m∈Z
η(m) δm
Autocorrelation coefficients
η(m) = limN→∞
1
2N+1
N∑
n=−Nw(n)w(n−m)
y Fourier coefficients of spectral measure σw = γ|[0,1)
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Diffraction versus dynamical spectrum
Dynamical system
(X,Z, µ) with Z ≃ {Tn | n ∈ Z}
y Hilbert space H = L2(X, µ)
y unitary operator on H,(UT f
)(x) := f(Tx)
y spectrum of UT (Koopman, von Neumann, Halmos)
Extension analogous definition for other groups, e.g. Rd
Spaces shifts, tilings, Delone sets, measures, ...
(Host 1986, Queffélec 1987, Pytheas Fogg 2002)
(Radin/Wolff 1992, Robinson 1996, Solomyak 1997)
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Diffraction versus dynamical spectrum
Theorem Let (X, Rd, µ) be an (ergodic) point set dynamicalsystem with diffraction γ. Then, γ is pure point iff (X, Rd, µ) has purepoint dynamical spectrum. The latter then is the group generatedby the support of γ, the so-called Fourier–Bohr spectrum of γ.
(Dworkin 1993, Hof 1995, Schlottmann 2000, Lee/Moody/Solomyak 2002, Baake/Lenz 2004,
Lenz/Strungaru 2009, Lenz/Moody 2012)
Theorem Let (X, Rd, µ) be as above, with finite local complexity.Then, the dynamical spectrum is fully determined by the diffractionmeasure of the system together with those of (part of) the family ofpatch-derived factors (patch locator sets).
(Miekisz/van Enter 1992, Baake/van Enter 2011, Baake/Lenz/van Enter 2013)
Remark Often, finitely many factors suffice !
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.5
Pure point spectra: Periodic crystals
Point measures: δx , δS :=∑
x∈S δx
Poisson summation formula: δΓ = dens(Γ ) δΓ ∗
for lattice Γ , dual lattice Γ ∗
Periodic crystals: ω = µ ∗ δΓ (µ finite)
⊲ γ = dens(Γ ) (µ ∗ µ) ∗ δΓ⊲ γ =
(dens(Γ )
)2 ∣∣µ∣∣2 δΓ ∗
(pure point: Bragg peaks on dual lattice)
⊲ dynamical spectrum Γ ∗ (pure point)
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Pure point spectra: Model sets
Silver mean substitution: a 7→ aba, b 7→ a (λPF = 1 +√
2 )
Fixed point: . . . abaaabaabaabaaaba|abaaabaabaabaaaba . . .
Silver mean point set: Λ ={x ∈ Z[
√2 ] | x′ ∈ [−
√2
2 ,√
22 ]
}
Lift to subset of lattice L ={(x, x′) | x ∈ Z[
√2 ]
}:
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.7
Pure point spectra: Model sets
CPS:
Rd π
←−−− Rd × R
mπint−−−−→ R
m
∪ ∪ ∪ dense
π(L)1−1←−−−− L −−−−→ π
int(L)
‖ ‖
L⋆
−−−−−−−−−−−−−−−−−−−−→ L⋆
Model set: Λ = {x ∈ L | x⋆ ∈ W } (assumed regular)
with W ⊂ Rm compact, λ(∂W ) = 0
Diffraction: γ =∑
k∈L⊛ |A(k)|2 δk (ω = δΛ) (pure point!)
with L⊛ = π(L∗) (Fourier module of Λ)
and amplitude A(k) = dens(Λ)vol(W ) 1W (−k⋆)
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.8
Example: Silver mean diffraction
L∗
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Example: Ammann–Beenker tiling
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Example: Ammann–Beenker tilingL = Z[ξ] L ∼ Z4 ⊂ R2 × R2 O: octagon
ξ = exp(2πi/8) φ(8) = 4 ⋆-map: ξ 7→ ξ3
ΛAB ={x ∈ Z1 + Zξ + Zξ2 + Zξ3 | x⋆ ∈ O
}
1
ξ
ξ2
ξ3
1⋆
ξ⋆
ξ2⋆
ξ3⋆
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.10
Example: Ammann–Beenker tiling
physical space internal space
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Example: Ammann–Beenker tilingDiffraction intensity:
Bragg peaks at positions k1 + k2ξ2 ∈ 1
2Z[ξ] ⊂ C
(where ξ = exp(2πi/8)) with intensities
I((k1, k2)
)=
1(4π2(k′2+k′1)(k
′2−k′1)
)2(
cos(k′2π) cos(λk′1π)
− cos(k′1π) cos(λk′2π)− k′1k′2
sin(k′2π) sin(λk′1π)
+k′2k′1
sin(k′1π) sin(λk′2π)
)2
with λ = 1 +√
2 and algebraic conjugation ′ :√
2 7→ −√
2
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Example: Ammann–Beenker tiling
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.10
Example: Visible pointsV =
{(x, y) ∈ Z2 | gcd(x, y) = 1
}= Z2/
⋃
p∈PpZ2
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Example: Visible points
γV =∑
k∈Q2 sq.f.
I(k) δk with I(k) =
(6
π2
∏
p|den(k)
1
1− p2
)2
(Baake, Grimm, Warrington 1994)
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.11
Example: Visible pointsModel set description:
L ={(x, ι(x)
)| x ∈ Z2
}⊂ Z2 ×H
with H =∏p∈P Z2/pZ2 compact group
⋆-map ι(x) = (xp)p∈P reduction of x mod p
window W =∏p∈P
(Z2/pZ2 \ {0}
)⊂ H
Window satisfies W = ∂W : weak model set(satisfies W ⊆ H relatively compact with θH(W ) > 0)(Baake, Huck, Strungaru 2017)
Note that the usual theory applies for regular model set with properwindows which satisfy ∅ 6= W = W ◦ and θH(∂W ) = 0, which canbe expanded to the case where θH(W ◦) = θH(W )
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Thue–Morse spectrum
Substitution: :1 7→ 11
1 7→ 11( 1 =−1 )
Iteration and fixed point:
1 7→ 11 7→ 1111 7→ 11111111 7→ . . . −→ v = (v) = v0v1v2v3 . . .
v2i = vi and v2i+1 = vi
⊲ weighted Dirac comb on Z with weights vi ∈ {±1}⊲ recursion for autocorrelation coefficients:
η(2m) = η(m) and η(2m+1) = −12
(η(m) + η(m+1)
)
for all m ∈ Z, with η(0) = 1
⊲ diffraction is purely singular continuous
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Thue–Morse spectrumAbsence of pp part:
Wiener’s criterion: γpp = 0 ⇐⇒ ΣN :=N∑
m=−N
(η(m)
)2= o(N)
Argument: Σ4N ≤ 32Σ2N (by recursion for η)
Absence of ac part:
Orthogonality of measures γac ⊥ γsc
⊲ split η(m) = ηac(m) + ηsc(m)
⊲ recursion holds for ηac separately, with ηac(0) free
Riemann-Lebesgue lemma: limm→±∞
ηac(m) = 0
⊲ ηac(0) = 0 ⊲ ηac(m) ≡ 0 (by recursion)
⊲ diffraction is purely singular continuous
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.12
Thue–Morse spectrumDistribution function F (x) = γ
([0, x]
)of diffraction measure
0 0.5 1
0
0.5
1
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Thue–Morse spectrumSliding block map: ψ : 11, 11 7→ a, 11, 11 7→ b
1111111111111111 1111111111111111
abaaabababaaabab abaaabababaaabaa
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Thue–Morse spectrumSliding block map: ψ : 11, 11 7→ a, 11, 11 7→ b
1111111111111111 1111111111111111
abaaabababaaabab abaaabababaaabaa
y period doubling: ′ :a 7→ ab
b 7→ aaXTM−−−→ XTM
ψ
yyψ (2:1)
Xpd′−−−→ Xpd
π
yyπ (a.e. 1:1)
Sol × 2−−−→ Sol
↑coincidence
⊲ model set(Dekking)
⊲ recovers pure point part Z[12 ] of dynamical spectrum
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.12
Thue–Morse spectrumResults generalise to bijective two-letter substitutionsincluding related systems (block substitutions) in higherdimensions
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.12
Rudin–Shapiro spectrum
Recursive definition:
w(−1) = −1, w(0) = 1, and
w(4n+ ℓ) =
{w(n), for ℓ ∈ {0, 1}(−1)n+ℓw(n), for ℓ ∈ {2, 3}
⊲ autocorrelation γRS = δ0
⊲ diffraction γRS = λ
⊲ diffraction purely absolutely continuous
⊲ homometric with fair coin tossing but zero entropy!
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.13
Rudin–Shapiro spectrum
Sliding block map ψ
1111111111111111 1111111111111111
bbaabaaabbabbaaa bbaabaabbbabbaaa
⊲ limit-periodic sequence
⊲ pure point diffraction
⊲ spectrum is supported on Z[12 ]
⊲ factor recovers pure point part of dynamical spectrum
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Random dimersRandom ‘dimers’ on Z (van Enter):
. . . [+ −][− +][− +][+ −][− +][− +][− +][+ −][+ −] . . .
. . . [− +][+ −][+ −][− +][+ −][+ −][+ −][− +][− +][+ −] . . .
⊲ shift space X, entropy 12 log(2)
⊲ stochastic Dirac comb with weights w(n) ∈ {±1}
⊲ autocorrelation γ = δ0 − 12(δ1 + δ−1) (a.s.)
⊲ diffraction γ =(1− cos(2πk)
)λ (a.s.)
⊲ ‘hidden’ order due to dimer constraints?
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.14
Random dimers
Sliding block map φ defined by (φw)(n) = −w(n)w(n+ 1)
⊲ maps X globally 2:1 to Y:
Y ={v ∈ {±1}Z | v(n) = 1 for all n ∈ 2Z or for all n ∈ 2Z + 1
}
⊲ signed Dirac comb vδZ
⊲ autocorrelation γ = 12δ0 + 1
2δ2Z
⊲ diffraction γ = 12λ+ 1
4δZ/2
⊲ uncovers ‘hidden’ pp part of dynamical spectrum
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.14
Summary & Outlook
Diffraction versus dynamical spectrum
Dynamical spectrum is richer
Diffraction as a useful tool
Patch-derived factors
‘Hidden’ order
Often finitely many factors suffice
Mixed spectra accessible
Extensions to higher dimensions
Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.15
ReferencesM. Baake & D. Lenz, Spectral notions of aperiodic order, Discr. Cont. Dyn. Syst. S 10(2017) 161–190
M. Baake, F. Gähler & U. Grimm, Spectral and topological properties of a family ofgeneralised Thue-Morse sequences, J. Math. Phys. 53 (2012) 032701
M. Baake, F. Gähler & U. Grimm, Examples of substitution systems and their factorsMichael Baake, J. Integer Sequences 16 (2013) 13.2.14
M. Baake & U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, CambridgeUniversity Press, Cambridge (2013)
M. Baake & U. Grimm, Squirals and beyond: Substitution tilings with singularcontinuous spectrum, Ergodic Th. & Dynam. Syst. 34 (2014) 1077–1102
M. Baake, C. Huck & N. Strungaru, On weak model sets of extremal density, Indag.Math. 28 (2017) 3–31
M. Baake, D. Lenz & A.C.D. van Enter, Dynamical versus diffraction spectrum forstructures with finite local complexity, Ergodic Th. & Dynam. Syst. 35 (2015)2017–2043
M. Baake & A.C.D. van Enter, Close-packed dimers on the line: Diffraction versusdynamical spectrum, J. Stat. Phys. 143 (2011) 88–101
A. Hof, On diffraction by aperiodic structures, Commun. Math. Phys. 169 (1995) 25–43
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