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Gap-labelling of the pinwheel tiling H. Moustafa Lab. de Math - - PowerPoint PPT Presentation

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Gap-labelling of the pinwheel tiling H. Moustafa Lab. de Math ematiques, Clermont-Ferrand France, CNRS UMR 6620 Vietri Sul Mare, August 31 2009 Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling

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SLIDE 1

Gap-labelling of the pinwheel tiling

  • H. Moustafa
  • Lab. de Math´

ematiques, Clermont-Ferrand France, CNRS UMR 6620

Vietri Sul Mare, August 31 2009

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SLIDE 2

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture Bellissard, 1989

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 4

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Connes, 1979 (Moore,Schochet, 1988)

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Connes, 1979 (Moore,Schochet, 1988) Douglas, Hurder and Kaminker, 1991

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Gap labelling for the pinwheel tiling

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Gap labelling for the pinwheel tiling Anderson and Putnam, 1998

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Gap labelling for the pinwheel tiling Anderson and Putnam, 1998 Bellissard and Savinien, 2007

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 10

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Gap labelling for the pinwheel tiling Computation

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Plan

Pinwheel tiling, tiling spaces and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Gap labelling for the pinwheel tiling Computation The gap-labelling is given by 1 264Z 1

5

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Pinwheel tiling, tiling spaces and the gap-labelling conjecture

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Definitions

Definition : Tiling of the plane : countable family P = {t1, t2, . . .} of non empty polygons ti, called tiles s.t. :

t1, t2, . . . cover the Euclidean plane. Two tiles only meet on their border.

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Definitions

Definition : Tiling of the plane : countable family P = {t1, t2, . . .} of non empty polygons ti, called tiles s.t. :

t1, t2, . . . cover the Euclidean plane. Two tiles only meet on their border.

Patch : finite union of tiles of the tiling.

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Pinwheel tiling

  • Fig. 1: Construction of a (1,2)-pinwheel tiling
  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Pinwheel tiling

  • Fig. 1: Construction of a (1,2)-pinwheel tiling
  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 17

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Pinwheel tiling

  • Fig. 1: Construction of a (1,2)-pinwheel tiling
  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 18

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Pinwheel tiling

  • Fig. 2: (1,2)-pinwheel tiling
  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Repetitivity

G = R2 ⋊ S1 group of rigid motions. Aperiodic tiling P : no translation of R2 fixes P. Finite G-type tiling : ∀ R > 0, there exists a finite number of patches with diameter smaller than R modulo the action of G. G-Repetitive tiling P : for any patch A of P, ∃R(A) > 0 s.t. any ball of radius R(A) intersects P on a patch containing a G-copy of A.

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Tiling space Ω

P a pinwheel tiling. Ω = completion of P · (R2 ⋊ S1). Ω is a compact metric space.

  • Ω, R2 ⋊ S1

is a minimal dynamical system. C(Ω) ⋊ R2 ⋊ S1 = completion of Cc(R2 ⋊ S1 × Ω).

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

The canonical transversal Ξ

Ξ := {P′ ∈ Ω | 0 ∈ Punct(P′) & P′ is well oriented}. Ξ is a Cantor set Ω is a foliated space and Ξ is a transversal of Ω.

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Gap-labelling conjecture

Ω is endowed with a G-invariant ergodic probability measure µ. µ induces an invariant transverse measure µt on Ξ defined locally by the quotient of µ by the Lebesgue measure . τ µ(f ) :=

  • Ω f (0, 0, ω)dµ(ω) for f ∈ Cc(R2 ⋊ S1 × Ω) defines

a trace on C(Ω) ⋊ R2 ⋊ S1.

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Tiling Construction Tiling space Ω The canonical transversal Ξ Gap-labelling conjecture

Gap-labelling conjecture

Gap-Labelling conjecture : ( Bellissard, 1989) τ µ

∗

  • K0
  • C(Ω) ⋊ R2 ⋊ S1

= µt C(Ξ, Z)

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Index theorem to solve the gap-labelling conjecture

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Index theorem to solve the gap-labelling conjecture

Theorem : (M., 2009) ∀ b ∈ K0

  • C(Ω) ⋊ R2 ⋊ S1

, ∃[u] ∈ K1

  • C(Ω)
  • s.t.

τ µ

∗

  • b
  • = τ µ

∗

  • [u] ⊗C(Ω) [D3]
  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Index theorem to solve the gap-labelling conjecture

K1 ` C(Ω) ´

⊗[D3]

  • K0

` C(Ω) ⋊ R2 ⋊ S1´

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Index theorem to solve the gap-labelling conjecture

K1 ` C(Ω) ´

⊗[d1]

  • ⊗[D3]
  • K0

` C(Ω) ⋊ S1´

⊗[D2]

K0

` C(Ω) ⋊ R2 ⋊ S1´

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Index theorem to solve the gap-labelling conjecture

ˇ H3(Ω; Z) ⊕ ˇ H1(Ω; Z) K1 ` C(Ω) ´

⊗[d1]

  • ⊗[D3]
  • ˇ

H2(Ω/S1; Z) ⊕ Z K0 ` C(Ω) ⋊ S1´

⊗[D2]

K0

` C(Ω) ⋊ R2 ⋊ S1´

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Index theorem to solve the gap-labelling conjecture

Theorem : (M., 2009) ∀ b ∈ K0

  • C(Ω) ⋊ R2 ⋊ S1

, ∃[u] ∈ K1

  • C(Ω)
  • s.t.

τ µ

∗

  • b
  • = τ µ

∗

  • [u] ⊗C(Ω) [D3]
  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Index theorem to solve the gap-labelling conjecture

Theorem : (M., 2009) ∀ b ∈ K0

  • C(Ω) ⋊ R2 ⋊ S1

, ∃[u] ∈ K1

  • C(Ω)
  • s.t.

τ µ

∗

  • b
  • = τ µ

∗

  • [u] ⊗C(Ω) [D3]
  • =
  • Ch3

ℓ([u]) | [Cµt]

  • where [Cµt] ∈ Hℓ

3(Ω) is the Ruelle-Sullivan current associated to µt

and Ch3

ℓ : K1(C(Ω)) → H3 ℓ (Ω) is the degree 3 component of the

longitudinal Chern character.

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Interger group of coinvariants

K1

  • C(Ω)
  • ch3

τ

  • ⊗C(Ω)
  • D3
  • H3

τ (Ω)

  • Cµt
  • K0
  • C(Ω) ⋊ R2 ⋊ S1

τ µ

∗

R

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Interger group of coinvariants

K1

  • C(Ω)
  • ch3

τ

  • ⊗C(Ω)
  • D3
  • ˇ

H3(Ω; Z)

r∗

  • H3

τ (Ω)

  • Cµt
  • K0
  • C(Ω) ⋊ R2 ⋊ S1

τ µ

∗

R

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Interger group of coinvariants

K1

  • C(Ω)
  • ch3

τ

  • ⊗C(Ω)
  • D3
  • ch3
  • ˇ

H3(Ω; Z)

r∗

  • H3

τ (Ω)

  • Cµt
  • K0
  • C(Ω) ⋊ R2 ⋊ S1

τ µ

∗

R

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Interger group of coinvariants

Theorem : ˇ H3(Ω; Z) ≃ H2(Ω/S1; Z) and Ω/S1 = lim

← − Bn

with Bn simplicial complexes of dimension 2. Thus ˇ H3(Ω; Z) ≃ lim

− →

ˇ H2(Bn; Z)

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Interger group of coinvariants

Theorem : ˇ H3(Ω; Z) ≃ H2(Ω/S1; Z) and Ω/S1 = lim

← − Bn

with Bn simplicial complexes of dimension 2. Thus ˇ H3(Ω; Z) ≃ lim

− →

ˇ H2(Bn; Z) ≃ lim

− → H2(Bn; Z)

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Interger group of coinvariants

Theorem : (M., 2009) lim

− → H2(Bn; Z) ≃ C(Ξ, Z)/H

with ∀h ∈ H, µt(h) = 0.

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 37

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Interger group of coinvariants

K1

  • C(Ω)
  • ch3

τ

  • ⊗C(Ω)
  • D3
  • ch3
  • ˇ

H3(Ω; Z)

r∗

  • C(Ξ, Z)/H
  • H3

τ (Ω)

  • Cµt
  • K0
  • C(Ω) ⋊ R2 ⋊ S1

τ µ

∗

R

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 38

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Interger group of coinvariants

K1

  • C(Ω)
  • ch3

τ

  • ⊗C(Ω)
  • D3
  • ch3
  • ˇ

H3(Ω; Z)

r∗

  • C(Ξ, Z)/H

µt

  • H3

τ (Ω)

  • Cµt
  • K0
  • C(Ω) ⋊ R2 ⋊ S1

τ µ

∗

R

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 39

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Index theorem Interger group of coinvariants

Interger group of coinvariants

Theorem : (M., 2009) τ µ

∗

  • K0
  • C(Ω) ⋊ R2 ⋊ S1

= µt C(Ξ, Z)

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 40

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Collared prototiles Computation

Computation

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Collared prototiles Computation

Definitions

P a pinwheel tiling. First corona of a tile : union of all the tiles intersecting it in P . collared prototile of P : equivalence class of tiles with the same first corona up to rigid motions.

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Collared prototiles Computation

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 43

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Collared prototiles Computation

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Collared prototiles Computation

Computation

A = matrix with ai,j = number of collared tiles of type i in the substitution of the collared prototile of type j. Proposition : (M. ,2009) lim

− →(Z108, A′) ≃ C(Ξ, Z)/H′

with ∀h ∈ H′ , µt(h) = 0. τ µ

∗

  • K0
  • C(Ω) ⋊ R2 ⋊ S1

= µt C(Ξ, Z)

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Collared prototiles Computation

Computation

A = matrix with ai,j = number of collared tiles of type i in the substitution of the collared prototile of type j. Proposition : (M. ,2009) lim

− →(Z108, A′) ≃ C(Ξ, Z)/H′

with ∀h ∈ H′ , µt(h) = 0. τ µ

∗

  • K0
  • C(Ω) ⋊ R2 ⋊ S1

= µt C(Ξ, Z)

  • = µt

lim

− →(Z108, A′)

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 46

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion Collared prototiles Computation

Computation

A = matrix with ai,j = number of collared tiles of type i in the substitution of the collared prototile of type j. Proposition : (M. ,2009) lim

− →(Z108, A′) ≃ C(Ξ, Z)/H′

with ∀h ∈ H′ , µt(h) = 0. τ µ

∗

  • K0
  • C(Ω) ⋊ R2 ⋊ S1

= µt C(Ξ, Z)

  • = µt

lim

− →(Z108, A′)

  • =

1 264Z 1 5

  • H. Moustafa

Gap-labelling of the pinwheel tiling

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SLIDE 47

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Computation Conclusion

Conclusion

  • H. Moustafa

Gap-labelling of the pinwheel tiling