Deformation of Delone Dynamical Systems and Pure Point Diffraction

This article deals with certain dynamical systems built from point sets and, moregenerally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystalsand aperiodic order, and important subclasses of them exhibit pure point diffractionspectra. We discuss the relevant framework and recall fundamental results and examples. Inparticular, we show that pure point diffraction is stable under equivariant local perturbationsand discuss various examples, including deformed model sets. A key step in the proof of stabilityconsists in transforming the problem into a question on factors of dynamical systems.