Related papers: Aperiodic order via dynamical systems: Diffraction…
We give a leisurely introduction into mathematical diffraction theory with a focus on pure point diffraction. In particular, we discuss various characterisations of pure point diffraction and common models arising from cut and project…
An extension of the Gutzwiller trace formula is given that includes diffraction effects due to hard wall scatterers or other singularities. The new trace formula involves periodic orbits which have arcs on the surface of singularity and…
We consider measurable and topological dynamical systems over locally compact abelian groups. Our main observation relates convergence of Wiener-Wintner type averages to eigenvalues of the dynamical system in question. As a consequence we…
We present a simplified model for dynamical diffraction of particles through a periodic thick perfect crystal based on repeated application of a coherent beam splitting unitary at coarse-grained lattice sites. By demanding translational…
Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of…
Multi-system interaction is an important and difficult problem in physics. Motivated by the experimental result of an electronic circuit element "Fractor", we introduce the concept of dynamic-order fractional dynamic system, in which the…
Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and…
Diffraction methods are at the heart of structure determination of solids. While Bragg-like scattering (pure point diffraction) is a characteristic feature of crystals and quasicrystals, it is not straightforward to interpret continuous…
We introduce a novel technique to find the asymptotic time behaviour of deterministic systems exhibiting anomalous diffusion. The procedure is tested for various classes of simple but physically relevant 1-D maps and possible relevance of…
Limit periodic point sets are aperiodic structures with pure point diffraction supported on a countably, but not finitely generated Fourier module that is based on a lattice and certain integer multiples of it. Examples are cut and project…
Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of…
Mathematical diffraction theory is concerned with the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra…
We study the problem of Arnold's diffusion in an example of isochronous system by using a geometrical method known as Windows Method. Despite the simple features of this example, we show that the absence of an anisochrony term leads to…
Diffraction methods are used to detect atomic order in solids. While uniquely ergodic systems with pure point diffraction have zero entropy, the relation between diffraction and entropy is not as straightforward in general. In particular,…
We study dynamical ordering of rods. In this process, rod alignment via pairwise interactions competes with diffusive wiggling. Under strong diffusion, the system is disordered, but at weak diffusion, the system is ordered. We present an…
This article considers the problem of diffraction by a wedge consisting of two semi-infinite periodic arrays of point scatterers. The solution is obtained in terms of two coupled systems, each of which is solved using the discrete…
This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them…
We present order reduction results for linear time invariant descriptor systems. Results are given for both forced and unforced systems as well methods for constructing the reduced order systems. Our results establish a precise connection…
We give a survey on classical and recent applications of dynamical systems to number theoretic problems. In particular, we focus on normal numbers, also including computational aspects. The main result is a sufficient condition for…
A simple model of 1D structure based on a Fibonacci sequence with variable atomic spacings is proposed. The model allows for observation of the continuous transition between periodic and non-periodic diffraction patterns. The diffraction…