Spaces of Tilings, Finite Telescopic Approximations and Gap-Labelling

For a large class of tilings of R, including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull ΩT of such a tiling T inherits a minimal R -lamination structure with flat leaves and a transversal ΓT which is a Cantor set. In this case, we show that the continuous hull can be seen as the projective limit of a suitable sequence of branched, oriented and flat compact d-manifolds. Truncated sequences furnish better and better finite approximations of the asymptotic dynamical system and the algebraic topological features related to this sequence reflect the dynamical properties of the R-action on the continuous hull. In particular the set of positive invariant measures of this action turns to be a convex cone, canonically associated with the orientation, in the projective limit of the d homology groups of the branched manifolds. As an application of this construction we prove a gap-labelling theorem: Consider the C-algebra AT of ΩT , and the group K0(AT ), then for every finite R-invariant measure μ on ΩT , the image of the group K0(AT ) by the μ-trace satisfies: