PV cohomology of pinwheel tilings, their integer group of coinvariants and gap-labelling

Abstract

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) and we then adapt the PV cohomology introduced in a paper of Bellissard and Savinien to define it for pinwheel tilings. We then prove that this cohomology is isomorphic to the integer Čech cohomology of the quotient of the hull by $S^1$ which let us prove that the top integer Čech cohomology of the hull is in fact the integer group of coinvariants on some transversal of the hull. The gap-labelling for pinwheel tilings is then proved and we end this article by an explicit computation of this gap-labelling, showing that $μ^t \big(C(Ξ,\ZZ) \big) = \dfrac{1}{264} \ZZ [\dfrac{1}{5}]$.

Problems of compilation by arxiv for figures on p.6 and p.7. I have only changed these figures

References in corpus (3)

• Pure Point Dynamical and Diffraction Spectra
• Conjugacies for Tiling Dynamical Systems
• About Substitution Tilings with Statistical Circular Symmetry

Cited by in corpus (3)

• Relating Diffraction and Spectral Data of Aperiodic Tilings: Towards a Bloch theorem
• K-theory of the Chair Tiling via AF-algebras
• New results on tilings via cup products and Chern characters on tiling spaces