Deformation of Delone dynamical systems and pure point diffraction

Abstract

This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them exhibit pure point diffraction spectra. We discuss the relevant framework and recall fundamental results and examples. In particular, we show that pure point diffraction is stable under ``equivariant'' local perturbations and discuss various examples,including deformed model sets. A key step in the proof of stability consists in transforming the problem into a question on factors of dynamical systems.

25 pages; revised version with minor corrections, an extended summary of the topic, and further references

References in corpus (3)

• Pure Point Dynamical and Diffraction Spectra
• When size matters: subshifts and their related tiling spaces
• Self-Similar Measures for Quasicrystals

Cited by in corpus (15)

• Pure Point spectrum for measure dynamical systems on locally compact Abelian groups
• Pure point diffraction and cut and project schemes for measures: The smooth case
• Dynamical versus diffraction spectrum for structures with finite local complexity
• Diffraction of stochastic point sets: Explicitly computable examples
• On the Fourier Analysis of Measures with Meyer Set Support
• Aperiodic order and pure point diffraction
• Dynamical properties of almost repetitive Delone sets
• Modulated crystals and almost periodic measures
• Close-packed dimers on the line: diffraction versus dynamical spectrum
• Random point sets and their diffraction
• Absence of singular continuous diffraction for discrete multi-component particle models
• On the Fourier Transformability of Strongly Almost Periodic Measures
• A note on measures vanishing at infinity
• Diffraction of the Hat and Spectre tilings and some of their relatives
• Deforming Meyer sets