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Ergodic Properties of Randomly Coloured Point Sets

Published online by Cambridge University Press:  20 November 2018

Peter Müller  and
Christoph Richard
Peter Müller
Affiliation:
Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraβe 39, 80333 München, Germany, e-mail: mueller@lmu.de
Christoph Richard
Affiliation:
Department für Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraβe 11, 91058 Erlangen, Germany, e-mail: richard@mi.uni-erlangen.de
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Abstract

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We provide a framework for studying randomly coloured point sets in a locally compact second-countable space on which a metrizable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical system for uniformly discrete uncoloured point sets. For point sets of finite local complexity, we characterize ergodicity geometrically in terms of pattern frequencies. The general framework allows us to incorporate a random colouring of the point sets. We derive an ergodic theorem for randomly coloured point sets with finite-range dependencies. Special attention is paid to the exclusion of exceptional instances for uniquely ergodic systems. The setup allows for a straightforward application to randomly coloured graphs

Keywords

37B5037A30Delone setsdynamical systems
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 2 , 01 April 2013 , pp. 349 - 402
Copyright
Copyright © Canadian Mathematical Society 2013

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