Wiener--Wintner points for topological dynamical systems
Daniel Lenz, Nicolae Strungaru
TL;DR
The paper develops a Wiener–Wintner type framework for measurable and topological dynamical systems over locally compact abelian groups, linking the existence of Wiener–Wintner averages to eigenvalues and Fourier–Bohr coefficients. It introduces Wiener–Wintner points and proves their prevalence in ergodic and metrizable topological settings, with deep ties to Besicovitch almost periodicity and pure point diffraction in aperiodic order. A general genericity-core approach underpins both the measurable and topological results, including uniform vanishing of Fourier–Bohr coefficients outside the eigenvalue spectrum. In the setting of translation bounded measures, the authors characterize Besicovitch almost periodicity as precisely the regime where the measure is Wiener–Wintner for some ergodic pure point system, connecting spectral properties to diffraction theory via the consistent phase property.
Abstract
We consider measurable and topological dynamical systems over locally compact abelian groups. Our main observation relates convergence of Wiener-Wintner type averages to eigenvalues of the dynamical system in question. As a consequence we infer existence of Fourier--Bohr coefficients for all characters for a set of points satisfying a specific genericity condition. In the topological case this leads naturally to the concept of what we call Wiener--Wintner point and we present a thorough study of such points. In particular we show that they have full measure in the ergodic case, and we relate them to Besicovitch almost periodicity. For dynamical systems of translation bounded measures, which are the crucial models in aperiodic order, our results give that the Wiener--Wintner points are exactly the points allowing for a diffraction theory with the consistent phase property.