Dynamics on the graph of the torus parametrisation

TL;DR

This paper investigates the dynamics of model sets derived from lattice projections on the torus, providing new insights into weak model sets and their spectral properties.

Contribution

It extends the torus-based approach to dynamics on the graph of the lattice-to-torus map, yielding new results on weak model sets and their spectral characteristics.

Findings

Proves pure point dynamical spectrum for the hull of weak model sets.

Derives a formula for pattern frequencies of maximal density configurations.

Provides transparent proofs of known results and new insights into weak model sets.

Abstract

Model sets are projections of certain lattice subsets. It was realised by Moody that dynamical properties of such sets are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map which associates lattice subsets to points of the torus and then transferring 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 together with the push forward of the torus Haar measure under the torus parametrisation map, and we derive a formula for the pattern frequencies of configurations with maximal density.