Spaces of tilings, finite telescopic approximations and gap-labelling

TL;DR

This paper studies tiling spaces with minimal lamination structures, showing they can be approximated by projective limits of branched manifolds, and applies this to prove a gap-labelling theorem.

Contribution

It introduces a novel approach to approximate tiling spaces using projective limits of branched manifolds and relates algebraic topology to dynamical properties.

Findings

Tiling spaces can be represented as projective limits of branched manifolds.

The set of invariant measures forms a convex cone in the projective limit.

A new gap-labelling theorem is established for these tiling spaces.

Abstract

For a large class of tilings, including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull of such a tiling inherits a minimal lamination structure with flat leaves and a transversal 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 manifolds.The algebraic topological features related to this sequence reflect the dynamical properties of the 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 top homology groups of the branched manifolds. As an application of this construction we prove a gap-labelling theorem.