Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra

Abstract

The paper investigates how correlations can completely specify a uniformly discrete point process. The setting is that of uniformly discrete point sets in real space for which the corresponding dynamical hull is ergodic. The first result is that all of the essential physical information in such a system is derivable from its $n$-point correlations, $n=2,3,>...$. If the system is pure point diffractive an upper bound on the number of correlations required can be derived from the cycle structure of a graph formed from the dynamical and Bragg spectra. In particular, if the diffraction has no extinctions, then the 2 and 3 point correlations contain all the relevant information.

16 pages

References in corpus (4)

• Pure Point Dynamical and Diffraction Spectra
• Homometric model sets and window covariograms
• Aperiodic order and pure point diffraction
• Dworkin's argument revisited: point processes, dynamics, diffraction, and correlations

Cited by in corpus (9)

• Symmetry Breaking and Order in the Age of Quasicrystals
• Mathematical diffraction of aperiodic structures
• Kinematic Diffraction from a Mathematical Viewpoint
• Dynamical versus diffraction spectrum for structures with finite local complexity
• Aperiodic crystals and beyond
• Aperiodic order and spherical diffraction, I: Auto-correlation of model sets
• Topological Bragg Peaks And How They Characterise Point Sets
• K-theory of crossed products of tiling C*-algebras by rotation groups
• How model sets can be determined by their two-point and three-point correlations