Dynamics on the graph of the torus parametrisation

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.

Some remarks were added, some proofs clarified, and Corollary 1 was extended in the revised version

References in corpus (7)

• Pure Point Dynamical and Diffraction Spectra
• Pure point diffraction and cut and project schemes for measures: The smooth case
• Ergodic properties of visible lattice points
• Pure point diffraction and Poisson summation
• Almost Periodic Measures and Meyer Sets
• On invariant measures for $\mathcal{B}$-free systems
• $\mathcal{B}$-free sets and dynamics

Cited by in corpus (22)

• Pure point diffraction and Poisson summation
• On the Fourier Analysis of Measures with Meyer Set Support
• Pure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity
• Aperiodic order and spherical diffraction, I: Auto-correlation of model sets
• A short guide to pure point diffraction in cut-and-project sets
• Dynamics of $\mathcal{B}$-free sets: a view through the window
• Periods and factors of weak model sets
• Modulated crystals and almost periodic measures
• Leptin densities in amenable groups
• Tautness for sets of multiples and applications to $\mathcal{B}$-free dynamics
• On arithmetic progressions in model sets
• Model sets with positive entropy in Euclidean cut and project schemes
• Pure point measures with sparse support and sparse Fourier--Bohr support
• Spectrum of weak model sets with Borel windows
• Model sets with precompact Borel windows
• A note on measures vanishing at infinity
• On the Garden of Eden theorem for B-free subshifts
• Why do Meyer sets diffract?
• Inter model sets in ${\mathbb{R}}^d$ are model sets
• Hereditary subshifts whose measure of maximal entropy has no Gibbs property
• Diffraction of the primes and other sets of zero density
• Fourier Transformable Measures with Meyer set support and their lift to the cut and project scheme