PV Cohomology of the Pinwheel Tilings, their Integer Group of Coinvariants and Gap-Labeling 15

In this paper, we first remind how we can see the hull of the pinwheel tiling as an inverse limit of simplicial complexes (Anderson and Putnam in Ergod Th Dynam Sys 18: 509-537, 1998) and we then adapt the PV cohomology introduced in Savinien and Bellissard (Ergod Th Dynam Sys 29: 997-1031, 2009) to define it for pinwheel tilings. We then prove that this cohomology is isomorphic to the integer. Cech cohomology of the quotient of the hull by S-1 which let us prove that the top integer Cech cohomology of the hull is in fact the integer group of coinvariants of the canonical transversal Xi of the hull. The gap-labeling for pinwheel tilings is then proved and we end this article by an explicit computation of this gap-labeling, showing that mu(t) (C(Xi, Z)) = 1/264 Z [1/5].