Eberlein decomposition for PV inflation systems

Abstract

The Dirac combs of primitive Pisot--Vijayaraghavan (PV) inflations on the real line or, more generally, in ${\mathbb{R}}^d$ are analysed. We construct a mean-orthogonal splitting for such Dirac combs that leads to the classic Eberlein decomposition on the level of the pair correlation measures, and thus to the separation of pure point versus continuous spectral components in the corresponding diffraction measures. This is illustrated with two guiding examples, and an extension to more general systems with randomness is outlined.

arXiv comment19 pages; revised version with minor improvements

References in corpus (17)

• Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra
• Pure point diffraction and cut and project schemes for measures: The smooth case
• Dense Dirac combs in Euclidean space with pure point diffraction
• Ergodic properties of randomly coloured point sets
• Diffraction of stochastic point sets: Explicitly computable examples
• Pure point diffraction and Poisson summation
• On the Fourier Analysis of Measures with Meyer Set Support
• Almost Periodic Measures and Meyer Sets
• Diffractive point sets with entropy
• Dynamical systems arising from random substitutions
• Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction
• On Weakly Almost Periodic Measures
• On Weighted Dirac Combs Supported Inside Model Sets
• Diffraction of Random Noble Means Words
• Three variations on a theme by Fibonacci
• Shifts of finite type and random substitutions
• Scaling properties of the Thue--Morse measure