Pure Point spectrum for measure dynamical systems on locally compact Abelian groups

Abstract

We show equivalence of pure point diffraction and pure point dynamical spectrum for measurable dynamical systems build from locally finite measures on locally compact Abelian groups. This generalizes all earlier results of this type. Our approach is based on a study of almost periodicity in a Hilbert space. It allows us to set up a perturbation theory for arbitrary equivariant measurable perturbations.

22 pages

References in corpus (3)

• Pure Point Dynamical and Diffraction Spectra
• Pure point diffraction and cut and project schemes for measures: The smooth case
• Aperiodic order and pure point diffraction

Cited by in corpus (19)

• Mathematical diffraction of aperiodic structures
• Kinematic Diffraction from a Mathematical Viewpoint
• 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 crystals and beyond
• Aperiodic order and spherical diffraction, I: Auto-correlation of model sets
• Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction
• On the Bragg Diffraction Spectra of a Meyer Set
• Periods and factors of weak model sets
• On norm almost periodic measures
• Modulated crystals and almost periodic measures
• Close-packed dimers on the line: diffraction versus dynamical spectrum
• Diffraction Theory of Point Processes: Systems with Clumping and Repulsion
• A note on measures vanishing at infinity
• On the Fourier Transformability of Strongly Almost Periodic Measures
• Diffraction of return time measures
• Why do Meyer sets diffract?
• Aperiodicity in equilibrium systems: Between order and disorder