Classification of disordered tilings
The notion of a disordered tiling is due to \textit{A. W. M. Dress} [`Relaxations of periodicity in the topological theory of tilings', J. Physique, Colloque C3, supplément au no 7, 47, C3-29--C3-40 (1986) and was studied by \textit{G. Brinkmann} [`Disorder in periodic tilings'. Preprint 91-068 SFB 343, Bielefeld (1991)]. It describes certain tilings which (locally) look like a periodic tiling almost everywhere. The precise definition is based on the Delaney-Dress symbol of an equivariant tiling and uses the chamber system (the set of maximal flags) associated with the tiling. This paper provides the following classification of these tilings: In \(\mathbf R^n\) (\(n\geq 3\)) (more generally: in simply connected manifolds without boundary which are simply connected at infinity) they are determined (up to almost isomorphy) by the minimal Delaney-Dress symbol, while in the plane the classification of disordered tilings needs the minimal Delaney-Dress symbol together with a conjugacy class of a cyclic subgroup of the symmetry group, a winding number, and (sometimes) the chirality.
scientific article; zbMATH DE number 1309785