14 research outputs found
Pure point diffraction and cut and project schemes for measures: The smooth case
We present cut and project formalism based on measures and continuous weight
functions of sufficiently fast decay. The emerging measures are strongly almost
periodic. The corresponding dynamical systems are compact groups and
homomorphic images of the underlying torus. In particular, they are strictly
ergodic with pure point spectrum and continuous eigenfunctions. Their
diffraction can be calculated explicitly. Our results cover and extend
corresponding earlier results on dense Dirac combs and continuous weight
functions with compact support. They also mark a clear difference in terms of
factor maps between the case of continuous and non-continuous weight functions.Comment: 30 page
Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra
The paper investigates how correlations can completely specify a uniformly
discrete point process. The setting is that of uniformly discrete point sets in
real space for which the corresponding dynamical hull is ergodic. The first
result is that all of the essential physical information in such a system is
derivable from its -point correlations, . If the system is
pure point diffractive an upper bound on the number of correlations required
can be derived from the cycle structure of a graph formed from the dynamical
and Bragg spectra. In particular, if the diffraction has no extinctions, then
the 2 and 3 point correlations contain all the relevant information.Comment: 16 page
How model sets can be determined by their two-point and three-point correlations
We show that real model sets with real internal spaces are determined, up to
translation and changes of density zero by their two- and three-point
correlations. We also show that there exist pairs of real (even one
dimensional) aperiodic model sets with internal spaces that are products of
real spaces and finite cyclic groups whose two- and three-point correlations
are identical but which are not related by either translation or inversion of
their windows. All these examples are pure point diffractive.
Placed in the context of ergodic uniformly discrete point processes, the
result is that real point processes of model sets based on real internal
windows are determined by their second and third moments.Comment: 19 page
Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
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.Comment: 16 pages; revised and slightly expanded versio
Aperiodic order and pure point 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 schemes. We finish with a list of open problems.Comment: 14 page
Discrete Hammersley's Lines with sources and sinks
International audienceWe introduce two stationary versions of two discrete variants of Hammersley's process in a finite box, this allows us to recover in a unified and simple way the laws of large numbers proved by T. Seppäläinen for two generalized Ulam's problems. As a by-product we obtain an elementary solution for the original Ulam problem. We also prove that for the first process defined on Z, Bernoulli product measures are the only extremal and translation-invariant stationary measures