Certain topological dynamical systems are considered that arise from actions of σ-compact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure point.

A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure. Simultaneously, we establish a natural link to the theory of the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model sets, but also to include examples such as the visible lattice points.

Abstract. We give an overview and extension of recent results on ergodic random Schrödinger operators for models on Zd. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable, at least locally. This limit, the integrated density of states (IDS), can be expressed by a closed Shubin-Pastur type trace formula. The set of points of increase of the IDS supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. This applies to several examples, including various periodic operators and percolation models. 1.

The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar random tilings based on solvable dimer models, augmented by a brief outline of the diraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous and absolutely continuous parts.

After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of lattice gases. We show that taking the site occupation of a model set stochastically results, with probabilistic certainty, in well-defined diffractive properties augmented by a constant diffuse background. We discuss both the case of independent, but identically distributed (i.i.d.) random variables and that of independent, but different (i.e., site dependent) random variables. Several examples are shown. Introduction Diffraction is one of the most important ways of identifying long-range order in mathematical and physical structures. In this paper, we look at the effects on diffraction that occur in certain periodic and quasiperiodic point sets when the occupation of the point sites is taken stochastically rather than deterministically, with in...

This article surveys the mathematics of the cut and project method as applied to point sets, called here model sets. It covers the geometric, arithmetic, and analytical sides of this theory as well as diffraction and the connection with dynamical systems. Dedicated to the memory of Richard (Dick) Slansky The spirit of the universe is subtle and informs all life. Things live and die and change their forms, without knowing the root from which they come. Abundantly it multiplies; eternally it stands by itself. The greatest reaches of space do not leave its confines, and the smallest down of a bird in autumn awaits its power to assume form. — Chuang Tzu (tr. Lin Yutang) 1

Abstract. We study randomly coloured graphs embedded into Euclidean space, whose vertex sets are infinite, uniformly discrete subsets of finite local complexity. We construct the appropriate ergodic dynamical systems, explicitly characterise ergodic measures, and prove an ergodic theorem. For covariant operators of finite range defined on those graphs, we show the existence and self-averaging of the integrated density of states, as well as the non-randomness of the spectrum. Our main result establishes Lifshits tails at the lower spectral edge of the graph Laplacian on bond percolation subgraphs, for sufficiently small probabilities. Among other assumptions, its proof requires exponential decay of the cluster-size distribution for percolation on rather general graphs. 1.

A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniform lattice Dirac comb, and its diraction measure is periodic, with the dual lattice as lattice of periods. This statement remains true in the setting of a locally compact Abelian group that is also -compact.

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Abstract. Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.