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Dynamics on the graph of the torus parametrization

Published online by Cambridge University Press:  19 September 2016

GERHARD KELLER  and
CHRISTOPH RICHARD
GERHARD KELLER
Affiliation:
Department Mathematik, Universität Erlangen-Nürnberg, Germany email [email protected]
CHRISTOPH RICHARD
Affiliation:
Department Mathematik, Universität Erlangen-Nürnberg, Germany email [email protected]
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Abstract

Model sets are projections of certain lattice subsets. It was realized by Moody [Uniform distribution in model sets. Canad. Math. Bull. 45(1) (2002), 123–130] that dynamical properties of such a set are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map that associates lattice subsets to points of the torus and then we transfer the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so-called weak model sets. In particular, we prove pure point dynamical spectrum for the hull of a weak model set of maximal density together with the push forward of the torus Haar measure under the torus parametrization map, and we derive a formula for its pattern frequencies.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 1048 - 1085
Copyright
© Cambridge University Press, 2016 

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