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Delone dynamical systems and spectral convergence

Part of: Classical measure theory Discrete geometry Selfadjoint operator algebras Locally compact groups and their algebras

Published online by Cambridge University Press:  22 October 2018

SIEGFRIED BECKUS  and
FELIX POGORZELSKI
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
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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.

Keywords

topological dynamicsgroup actionsoperator algebrasHamiltonian dynamics

MSC classification

Primary: 28A33: Spaces of measures, convergence of measures 22D40: Ergodic theory on groups 52C23: Quasicrystals, aperiodic tilings
Secondary: 46L60: Applications of selfadjoint operator algebras to physics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1510 - 1544
Copyright
© Cambridge University Press, 2018

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