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Delone dynamical systems and spectral convergence

Published online by Cambridge University Press:  22 October 2018

SIEGFRIED BECKUS
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel email beckus.siegf@technion.ac.il
FELIX POGORZELSKI
Affiliation:
Department of Mathematics, University of Leipzig, 04109 Leipzig, Germany email felix.pogorzelski@math.uni-leipzig.de

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.

Type
Original Article
Copyright
© Cambridge University Press, 2018

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