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Dynamical versus diffraction spectrum for structures with finite local complexity

Published online by Cambridge University Press:  05 August 2014

MICHAEL BAAKE ,
DANIEL LENZ  and
AERNOUT VAN ENTER
MICHAEL BAAKE
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany email mbaake@math.uni-bielefeld.de
DANIEL LENZ
Affiliation:
Fakultät für Mathematik, Universität Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany email daniel.lenz@uni-jena.de
AERNOUT VAN ENTER
Affiliation:
Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands email a.c.d.van.enter@rug.nl
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Abstract

It is well known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynamical spectrum with a collection of diffraction spectra of the system and certain factors. This equivalence gives access to the dynamical spectrum via these diffraction spectra. It is particularly useful as the diffraction spectra are often simpler to determine and, in many cases, only very few of them need to be calculated.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015 , pp. 2017 - 2043
Copyright
© Cambridge University Press, 2014 

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