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Ergodic Properties of Randomly Coloured Point Sets

  Published:2012-05-10
 Printed: Apr 2013
  • Peter Müller,
    Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333 München, Germany
  • Christoph Richard,
    Department für Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany
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Abstract

We provide a framework for studying randomly coloured point sets in a locally compact, second-countable space on which a metrisable 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 characterise ergodicity geometrically in terms of pattern frequencies. The general framework allows 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: Delone sets, dynamical systems Delone sets, dynamical systems
MSC Classifications: 37B50, 37A30 show english descriptions Multi-dimensional shifts of finite type, tiling dynamics
Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35}
37B50 - Multi-dimensional shifts of finite type, tiling dynamics
37A30 - Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35}
 

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