Spaces of tilings, finite telescopic approximations and gap-labeling

The continuous Hull of a repetitive tiling T in ℝd with the Finite Pattern Condition (FPC) inherits a minimal ℝd-lamination structure with flat leaves and a transversal [inline-graphic not available: see fulltext] 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 ℝd-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 [inline-graphic not available: see fulltext] 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 dth homology group canonically associated with the orientation of ℝd. At last, the gap labeling theorem holds: given an invariant ergodic probability measure μ on the Hull the corresponding Integrated Density of States (IDS) of any selfadjoint operators affiliated to [inline-graphic not available: see fulltext] takes on values on spectral gaps in the ℤ-module generated by the occurrence probabilities of finite patches in the tiling.