Journal Article10.1017/S0143385797084988
Dynamics of self-similar tilings
TL;DR: The main focus of as mentioned in this paper is on spectral properties of self-similar and self-affine tilings, which are shown to be uniquely ergodic in terms of weak mixing and pure discrete spectrum.
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Abstract: This paper investigates dynamical systems arising from the
action by
translations on the orbit closures of self-similar and
self-affine tilings
of . The main focus is on spectral properties of such
systems
which are shown to be uniquely ergodic. We establish
criteria for
weak mixing and pure discrete spectrum for wide classes of
such systems.
They are applied to a number of examples which include
tilings with
polygonal and fractal tile boundaries; systems with pure
discrete,
continuous and mixed spectrum.
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Citations
Processes on Unimodular Random Networks
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TL;DR: In this article, the authors investigate unimodular random networks and their properties via reversibility of an associated random walk and their similarities to unimmodular quasi-transitive graphs, and extend various theorems concerning random walks, percolation, spanning forests, and amenability.
Topological invariants for substitution tilings and their associated $C^\ast$-algebras
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TL;DR: In this article, it was shown that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids.
337
•Posted Content
Processes on Unimodular Random Networks
David Aldous,Russell Lyons +1 more
TL;DR: In this article, the authors investigate unimodular random networks and their properties via reversibility of an associated random walk and their similarities to unimmodular quasi-transitive graphs, and extend various theorems concerning random walks, percolation, spanning forests, and amenability.
302
Model Sets: A Survey
TL;DR: Even when reduced to its simplest form, namely that of point sets in euclidean space, the phenomenon of genuine quasi-periodicity appears extraordinary as discussed by the authors Although it seems unfruitful to try and define the concept precisely, the following properties may be considered as representative.
240
Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra
Michael Baake,Daniel Lenz +1 more
TL;DR: In this article, the authors consider topological dynamical systems that arise from locally compact Abelian groups on compact spaces of translation bounded measures and show that such a system has a pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
229
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Quasicrystals and geometry
Marjorie Senechal
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TL;DR: An atlas of tiling transforms is given in this article, along with a mathematical toolbag for tiling transform analysis and a diagram of the Penrose tilings of the plane.