164 research outputs found
Unique continuation for the magnetic Schrödinger equation
The unique‐continuation property from sets of positive measure is here proven for the many‐body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one‐body or two‐body functions, typical for Hamiltonians in many‐body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique‐continuation property plays an important role in density‐functional theories, which underpins its relevance in quantum chemistry
A test for a conjecture on the nature of attractors for smooth dynamical systems
Dynamics arising persistently in smooth dynamical systems ranges from regular
dynamics (periodic, quasiperiodic) to strongly chaotic dynamics (Anosov,
uniformly hyperbolic, nonuniformly hyperbolic modelled by Young towers). The
latter include many classical examples such as Lorenz and H\'enon-like
attractors and enjoy strong statistical properties.
It is natural to conjecture (or at least hope) that most dynamical systems
fall into these two extreme situations. We describe a numerical test for such a
conjecture/hope and apply this to the logistic map where the conjecture holds
by a theorem of Lyubich, and to the Lorenz-96 system in 40 dimensions where
there is no rigorous theory. The numerical outcome is almost identical for both
(except for the amount of data required) and provides evidence for the validity
of the conjecture.Comment: Accepted version. Minor modifications from previous versio
Large deviations for non-uniformly expanding maps
We obtain large deviation results for non-uniformly expanding maps with
non-flat singularities or criticalities and for partially hyperbolic
non-uniformly expanding attracting sets. That is, given a continuous function
we consider its space average with respect to a physical measure and compare
this with the time averages along orbits of the map, showing that the Lebesgue
measure of the set of points whose time averages stay away from the space
average decays to zero exponentially fast with the number of iterates involved.
As easy by-products we deduce escape rates from subsets of the basins of
physical measures for these types of maps. The rates of decay are naturally
related to the metric entropy and pressure function of the system with respect
to a family of equilibrium states. The corrections added to the published
version of this text appear in bold; see last section for a list of changesComment: 36 pages, 1 figure. After many PhD students and colleagues having
pointed several errors in the statements and proofs, this is a correction to
published article answering those comments. List of main changes in a new
last sectio
Statistical stability and continuity of SRB entropy for systems with Gibbs-Markov structures
We present conditions on families of diffeomorphisms that guarantee
statistical stability and SRB entropy continuity. They rely on the existence of
horseshoe-like sets with infinitely many branches and variable return times. As
an application we consider the family of Henon maps within the set of
Benedicks-Carleson parameters
Exponential speed of mixing for skew-products with singularities
Let be the
endomorphism given by where is a positive real number. We prove that is
topologically mixing and if then is mixing with respect to Lebesgue
measure. Furthermore we prove that the speed of mixing is exponential.Comment: 23 pages, 3 figure
From limit cycles to strange attractors
We define a quantitative notion of shear for limit cycles of flows. We prove
that strange attractors and SRB measures emerge when systems exhibiting limit
cycles with sufficient shear are subjected to periodic pulsatile drives. The
strange attractors possess a number of precisely-defined dynamical properties
that together imply chaos that is both sustained in time and physically
observable.Comment: 27 page
Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems
In this paper, we prove an inequality, which we call "Devroye inequality",
for a large class of non-uniformly hyperbolic dynamical systems (M,f). This
class, introduced by L.-S. Young, includes families of piece-wise hyperbolic
maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas),
unimodal and H{\'e}non-like maps. Devroye inequality provides an upper bound
for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K
is any separately Holder continuous function of n variables. In particular, we
can deal with observables which are not Birkhoff averages. We will show in
\cite{CCS} some applications of Devroye inequality to statistical properties of
this class of dynamical systems.Comment: Corrected version; To appear in Nonlinearit
Infinitely Many Stochastically Stable Attractors
Let f be a diffeomorphism of a compact finite dimensional boundaryless
manifold M exhibiting infinitely many coexisting attractors. Assume that each
attractor supports a stochastically stable probability measure and that the
union of the basins of attraction of each attractor covers Lebesgue almost all
points of M. We prove that the time averages of almost all orbits under random
perturbations are given by a finite number of probability measures. Moreover
these probability measures are close to the probability measures supported by
the attractors when the perturbations are close to the original map f.Comment: 14 pages, 2 figure
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