Voronoi Tessellations for Matchbox Manifolds

Matchbox manifolds M are a special class of foliated spaces, which include as special examples the weak solenoids, suspensions of odometer and Toeplitz actions, and tiling spaces associated to aperiodic tilings with finite local complexity. They have many properties analogous to those of a compact manifold, but the additional data inherently encoded in the pseudogroup dynamics of its foliation F represent fundamental groupoid data. As such, they are a rich class of mathematical objects to study. The special cases of weak solenoids, suspensions of group actions, and tiling spaces have an additional structure, that of a transverse foliation, consisting of a continuous family of Cantor sets transverse to the foliated structure. The purpose of this paper is to show that this transverse structure can be defined on all equicontinuous matchbox manifolds, as well as on special foliated subsets of general matchbox manifolds. This follows from the construction of uniform Voronoi tessellations on leaves, which is the main goal of this work. From this, we define a foliated Delaunay triangulation of M, adapted to the dynamics of F . The result is highly technical, but underlies the study of the basic topological structure of matchbox manifolds in general. Our methods are similar to some prior results in the literature [35, 43], though are unique in that we give the construction of the Voronoi tessellations for a complete Riemannian manifold L of arbitrary dimension, while the constructions in the literature apply only to the case where the manifold L is Euclidean space Rn.