Delone dynamical systems and spectral convergence

TL;DR

This paper investigates the spectral and autocorrelation properties of Delone sets within locally compact groups, establishing continuity results and approximation methods for spectral measures in Euclidean spaces.

Contribution

It introduces a dynamical systems framework for analyzing spectral convergence and autocorrelation in Delone sets, extending understanding of spectral limits and approximations.

Findings

Spectral measures are continuous under Chabauty-Fell convergence for uniquely ergodic systems.

Measured quantities can be approximated by periodic analogs in Euclidean spaces.

The approach links dynamical systems with spectral analysis of point sets.

Abstract

In the realm of Delone sets in locally compact, second countable, Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty-Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.