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

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 conjecture Computation Conclusion Plan Pinwheel tiling, tiling spaces and the gap labelling conjecture H. Moustafa Gap-labelling of the pinwheel tiling

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

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

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

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

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

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

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

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

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 1 � 1 � The gap-labelling is given by 264 Z 5 H. Moustafa Gap-labelling of the pinwheel tiling

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Pinwheel tiling, tiling spaces and the gap-labelling conjecture H. Moustafa Gap-labelling of the pinwheel tiling

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Definitions Definition : Tiling of the plane : countable family P = { t 1 , t 2 , . . . } of non empty polygons t i , called tiles s.t. : t 1 , t 2 , . . . cover the Euclidean plane. Two tiles only meet on their border. H. Moustafa Gap-labelling of the pinwheel tiling

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Definitions Definition : Tiling of the plane : countable family P = { t 1 , t 2 , . . . } of non empty polygons t i , called tiles s.t. : t 1 , t 2 , . . . 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

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Pinwheel tiling Fig. 1: Construction of a (1,2)-pinwheel tiling H. Moustafa Gap-labelling of the pinwheel tiling

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Pinwheel tiling Fig. 1: Construction of a (1,2)-pinwheel tiling H. Moustafa Gap-labelling of the pinwheel tiling

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Pinwheel tiling Fig. 1: Construction of a (1,2)-pinwheel tiling H. Moustafa Gap-labelling of the pinwheel tiling

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Pinwheel tiling Fig. 2: (1,2)-pinwheel tiling H. Moustafa Gap-labelling of the pinwheel tiling

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Repetitivity G = R 2 ⋊ S 1 group of rigid motions. Aperiodic tiling P : no translation of R 2 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

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Tiling space Ω P a pinwheel tiling. Ω = completion of P · ( R 2 ⋊ S 1 ). Ω is a compact metric space. Ω , R 2 ⋊ S 1 � � is a minimal dynamical system. C (Ω) ⋊ R 2 ⋊ S 1 = completion of C c ( R 2 ⋊ S 1 × Ω). H. Moustafa Gap-labelling of the pinwheel tiling

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

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Gap-labelling conjecture Ω is endowed with a G -invariant ergodic probability measure µ . µ induces an invariant transverse measure µ t on Ξ defined locally by the quotient of µ by the Lebesgue measure . Ω f (0 , 0 , ω ) d µ ( ω ) for f ∈ C c ( R 2 ⋊ S 1 × Ω) defines τ µ ( f ) := � a trace on C (Ω) ⋊ R 2 ⋊ S 1 . H. Moustafa Gap-labelling of the pinwheel tiling

Tiling Pinwheel tiling and the gap labelling conjecture Construction Index theorem to solve the gap-labelling conjecture Tiling space Ω Computation The canonical transversal Ξ Conclusion Gap-labelling conjecture Gap-labelling conjecture Gap-Labelling conjecture : ( Bellissard, 1989) � C (Ω) ⋊ R 2 ⋊ S 1 �� τ µ = µ t � � � K 0 C (Ξ , Z ) ∗ H. Moustafa Gap-labelling of the pinwheel tiling

Pinwheel tiling and the gap labelling conjecture Index theorem to solve the gap-labelling conjecture Index theorem Computation Interger group of coinvariants Conclusion Index theorem to solve the gap-labelling conjecture H. Moustafa Gap-labelling of the pinwheel tiling