1. CJM 2012 (vol 65 pp. 349)
||Ergodic Properties of Randomly Coloured Point Sets|
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
Keywords:Delone sets, dynamical systems
2. CJM 2011 (vol 65 pp. 149)
||Equicontinuous Delone Dynamical Systems|
We characterize equicontinuous Delone dynamical systems as those
coming from Delone sets with strongly almost periodic Dirac combs.
Within the class of systems with finite local complexity, the only
equicontinuous systems are then shown to be the crystallographic
ones. On the other hand, within the class without finite local
complexity, we exhibit examples of equicontinuous minimal Delone
dynamical systems that are not crystallographic.
Our results solve the problem posed by Lagarias as to whether a Delone
set whose Dirac comb is strongly almost periodic must be
Keywords:Delone sets, tilings, diffraction, topological dynamical systems, almost periodic systems