Spaces of tilings, finite telescopic approximations and gap-labeling

The continuous Hull of a repetitive tiling T in R-d with the Finite Pattern Condition (FPC) inherits a minimal R-d-lamination structure with flat leaves and a transversal Gamma(T) which is a Cantor set. This class of tiling includes the Penrose & the Amman Benkker ones in 2D, as well as the icosahedral tilings in 3D. We show that the continuous Hull, with its canonical R-d-action, can be seen as the projective limit of a suitable sequence of branched, oriented and flat compact d-manifolds. As a consequence, the longitudinal cohomology and the K-theory of the corresponding C*-algebra A(T) are obtained as direct limits of cohomology and K-theory of ordinary manifolds. Moreover, the space of invariant finite positive measures can be identified with a cone in the d(th) homology group canonically associated with the orientation of R-d. At last, the gap labeling theorem holds: given an invariant ergodic probability measure mu on the Hull the corresponding Integrated Density of States (IDS) of any selfadjoint operators affiliated to A(T) takes on values on spectral gaps in the Z-module generated by the occurrence probabilities of finite patches in the tiling.