2015/11/30 by Gerhard Keller, Christoph Richard · 3 citations
Computer Science · Mathematics · #Cellular Automata and Applications #Combinatorics #Discrete mathematics #Dynamical systems theory #Geometric and Algebraic Topology #Geometry #Graph #Lattice (music) #Mathematical Dynamics and Fractals #Mathematical proof #Mathematics #Physics #Statistical physics #Torus #math.DS #msc:37A25 #msc:37B10 #msc:37B50 #msc:52C23
paper · pdf · doi:10.1017/etds.2016.53
Some remarks were added, some proofs clarified, and Corollary 1 was extended in the revised version
arxiv created 2016/05/10 · openalex created_date 2016/06/24 · openalex publication_date 2016/09/19 · arxiv updated 2018/03/01 · openalex updated_date 2026/08/05
Model sets are projections of certain lattice subsets. It was realized by Moody [Uniform distribution in model sets. Canad. Math. Bull. 45 (1) (2002), 123–130] that dynamical properties of such a set are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map that associates lattice subsets to points of the torus and then we transfer the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so-called weak model sets. In particular, we prove pure point dynamical spectrum for the hull of a weak model set of maximal density together with the push forward of the torus Haar measure under the torus parametrization map, and we derive a formula for its pattern frequencies.