Ergodic properties of randomly coloured point sets
arXiv:1005.4884 · doi:10.4153/CJM-2012-009-7
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.
This version is almost identical to the version published electronically on May 10, 2012 in the Canadian Journal of Mathematics
References in corpus (1)
Cited by in corpus (18)
- Dynamics on the graph of the torus parametrisation
- On pattern entropy of weak model sets
- Dynamical versus diffraction spectrum for structures with finite local complexity
- Dynamical properties of almost repetitive Delone sets
- Characterization of the Anderson metal-insulator transition for non ergodic operators and application
- Ergodicity and dynamical localization for Delone-Anderson operators
- Leptin densities in amenable groups
- Localisation for Delone operators via Bernoulli randomisation
- On sampling and interpolation by model sets
- Convergence theorems for graph sequences
- Spectrum of weak model sets with Borel windows
- Linear repetitivity beyond abelian groups
- Random Schrödinger Operators and Anderson localization in aperiodic media
- Substitution tilings with dense tile orientations and n-fold rotational symmetry
- Eberlein decomposition for PV inflation systems
- Diffraction of return time measures
- Random Schrödinger Operators on discrete structures
- Random fields on model sets with localized dependency and their diffraction