Spaces of Tilings, Finite Telescopic Approximations and Gap-Labeling

Abstract

For a large class of tilings of R d , including the Penrose tiling in dimension 2 as well as the icosahedral ones in dimension 3, the continuous hull Ω T of such a tiling T inherits a minimal R d -lamination structure with flat leaves and a transversal Γ T 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 d -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 th -homology groups of the branched manifolds. As an application of this construction we prove a gap-labelling theorem:Consider the C * -algebra A T of Ω T , and the group K 0 (A T ), then for every finite R d -invariant measure µ on Ω T , the image of the group K 0 (A T ) by the µ-trace satisfies:where µ t is the transverse invariant measure on Γ T induced by µ and C(Γ T , Z) is the set of continuous functions on Γ T with integer values.

A spectral sequence for theK-theory of tiling spaces

“…These two results are sufficient to get the K-theory of the hull which, thanks to the Thom-Connes theorem [21], gives also the K-theory of the C * -algebra of the tiling C(Ω) ⋊ R d . However, the construction of the hull through an inverse limit of branched manifolds, initiated by Anderson and Putnam [1] for the case of substitution tilings and generalized in [13] to all repetitive tilings with finite local complexity, suggests a different and more canonical construction. So far however, it is not yet efficient for practical calculations.…”

“…The present paper generalizes this construction for tilings by replacing the classifying space T d , by the prototile space B 0 . The hull can also be built out of a box decomposition [13]. Namely a box is a local product of the transversal (which is a Cantor set) by a polyhedron in R d called the base of the box.…”

“…Then Kasparov [39,40] defined the notion of KK-theory generalizing even more the K groups to correspondences between two C * -algebras. The problem investigated in the present paper is the latest development of a program that was initiated in the early eighties [38,5] when the first version of the gap labeling theorem was proved (see [6,9,13] for later developments). At that time the problem was to compute the spectrum of a Schrödinger operator H in an aperiodic potential.…”

“…The hull of a tiling is seen as a dynamical system (Ω, R d , t) which, for the class of tilings considered here, has interesting properties that are now stated (see [13] …”

“…Although those results will not be used here, it has been proven in [13] that the prototile space B 0 (as well as the patch spaces B p defined similarly in the next section, definition 10) has the structure of a flat oriented Riemanian branched manifold. Also, the hull Ω can be given a lamination structure as follows: the lifts of the interiors of the tiles B 0j = p −1 0 ( • τ j ) are boxes of the lamination which are homeomorphic to • t j ×Ξ(τ j ) via the maps (x, ξ) → t −x ξ which read as local charts.…”