Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe...

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Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.1

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Page 1: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 2: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Perspective

Harmonic Analysis Dynamical Systems

ց ւ

Alg. / Comb. → Aperiodic Order ← Topology

ր տ

Number Theory Discrete Geometry

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.2

Page 3: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Origins in crystallography

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.3

Page 4: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Origins in crystallography

(Photo courtesy of the Ames Laboratory)

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.3

Page 5: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.3

Page 6: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Origins in crystallography

Wolf Prize in Physics 1999Nobel Prize in Chemistry 2011

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.3

Page 7: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.4

Page 8: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 9: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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)

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.4

Page 10: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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)

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.5

Page 11: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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 !

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Page 12: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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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Page 13: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 14: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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⋆)

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Page 15: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Example: Silver mean diffraction

L∗

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Page 16: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Example: Ammann–Beenker tiling

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.10

Page 17: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 18: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Example: Ammann–Beenker tiling

physical space internal space

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.10

Page 19: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.10

Page 20: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Example: Ammann–Beenker tiling

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.10

Page 21: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Example: Visible pointsV =

{(x, y) ∈ Z2 | gcd(x, y) = 1

}= Z2/

p∈PpZ2

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.11

Page 22: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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)

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Page 23: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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 )

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.11

Page 24: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.12

Page 25: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 26: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Thue–Morse spectrumDistribution function F (x) = γ

([0, x]

)of diffraction measure

0 0.5 1

0

0.5

1

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.12

Page 27: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

Thue–Morse spectrumSliding block map: ψ : 11, 11 7→ a, 11, 11 7→ b

1111111111111111 1111111111111111

abaaabababaaabab abaaabababaaabaa

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.12

Page 28: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 29: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 30: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 31: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Workshop on Aperiodicity and Hierarchical Structures in Tilings, Lyon, 25–29 September 2017 – p.13

Page 32: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 33: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 34: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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

Page 35: Spectral Properties of Aperiodic Structures · Spectral Properties of Aperiodic Structures Uwe Grimm School of Mathematics & Statistics The Open University Milton Keynes, UK Workshop

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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