64,886 research outputs found
Diffraction of return time measures
Letting denote an ergodic transformation of the unit interval and letting
denote an observable, we construct the
-weighted return time measure for a reference point as
the weighted Dirac comb with support in and weights at , and if is non-invertible, then we set the
weights equal to zero for all . Given such a Dirac comb, we are
interested in its diffraction spectrum which emerges from the Fourier transform
of its autocorrelation and analyse it for the dependence on the underlying
transformation. For certain rapidly mixing transformations and observables of
bounded variation, we show that the diffraction of consists of a
trivial atom and an absolutely continuous part, almost surely with respect to
. This contrasts what occurs in the setting of regular model sets arising
from cut and project schemes and deterministic incommensurate structures. As a
prominent example of non-mixing transformations, we consider the family of
rigid rotations with rotation
number . In contrast to when is mixing, we observe
that the diffraction of is pure point, almost surely with respect to
. Moreover, if is irrational and the observable is Riemann
integrable, then the diffraction of is independent of . Finally,
for a converging sequence of rotation
numbers, we provide new results concerning the limiting behaviour of the
associated diffractions.Comment: 11 pages, 2 figure
Dynamical versus diffraction spectrum for structures with finite local complexity
It is well-known that the dynamical spectrum of an ergodic measure dynamical
system is related to the diffraction measure of a typical element of the
system. This situation includes ergodic subshifts from symbolic dynamics as
well as ergodic Delone dynamical systems, both via suitable embeddings. The
connection is rather well understood when the spectrum is pure point, where the
two spectral notions are essentially equivalent. In general, however, the
dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of
finite local complexity and establish the equivalence of the dynamical spectrum
with a collection of diffraction spectra of the system and certain factors.
This equivalence gives access to the dynamical spectrum via these diffraction
spectra. It is particularly useful as the diffraction spectra are often simpler
to determine and, in many cases, only very few of them need to be calculated.Comment: 27 pages; some minor revisions and improvement
Metrics and spectral triples for Dirichlet and resistance forms
The article deals with intrinsic metrics, Dirac operators and spectral
triples induced by regular Dirichlet and resistance forms. We show, in
particular, that if a local resistance form is given and the space is compact
in resistance metric, then the intrinsic metric yields a geodesic space. Given
a regular Dirichlet form, we consider Dirac operators within the framework of
differential 1-forms proposed by Cipriani and Sauvageot, and comment on its
spectral properties. If the Dirichlet form admits a carr\'e operator and the
generator has discrete spectrum, then we can construct a related spectral
triple, and in the compact and strongly local case the associated Connes
distance coincides with the intrinsic metric. We finally give a description of
the intrinsic metric in terms of vector fields
A pure jump Markov process with a random singularity spectrum
We construct a non-decreasing pure jump Markov process, whose jump measure
heavily depends on the values taken by the process. We determine the
singularity spectrum of this process, which turns out to be random and to
depend locally on the values taken by the process. The result relies on fine
properties of the distribution of Poisson point processes and on ubiquity
theorems.Comment: 20 pages, 4 figure
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