Wiener--Wintner points for topological dynamical systems

TL;DR

This paper investigates Wiener--Wintner points in topological dynamical systems, establishing their connection to eigenvalues, Fourier--Bohr coefficients, and aperiodic order, with results on their prevalence and properties.

Contribution

It introduces the concept of Wiener--Wintner points in topological systems and explores their properties, including measure-theoretic and spectral aspects, linking to aperiodic order and diffraction theory.

Findings

Wiener--Wintner points have full measure in ergodic systems.

They are characterized by the existence of Fourier--Bohr coefficients for all characters.

Wiener--Wintner points correspond to points allowing diffraction with the 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.