3 research outputs found
Close-packed dimers on the line: diffraction versus dynamical spectrum
The translation action of \RR^{d} on a translation bounded measure
leads to an interesting class of dynamical systems, with a rather rich spectral
theory. In general, the diffraction spectrum of , which is the carrier
of the diffraction measure, live on a subset of the dynamical spectrum. It is
known that, under some mild assumptions, a pure point diffraction spectrum
implies a pure point dynamical spectrum (the opposite implication always being
true). For other systems, the diffraction spectrum can be a proper subset of
the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with
singular continuous diffraction) in \cite{EM}. Here, we construct a random
system of close-packed dimers on the line that have some underlying long-range
periodic order as well, and display the same type of phenomenon for a system
with absolutely continuous spectrum. An interpretation in terms of `atomic'
versus `molecular' spectrum suggests a way to come to a more general
correspondence between these two types of spectra.Comment: 14 pages, with some additions and improvement
Diffraction of stochastic point sets: Explicitly computable examples
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 reasonable generality. The latter is based on the classical theory of point processes and the Palm distribution. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the Poisson summation formula for lattice Dirac combs