356 research outputs found
Statistical stability and limit laws for Rovella maps
We consider the family of one-dimensional maps arising from the contracting
Lorenz attractors studied by Rovella. Benedicks-Carleson techniques were used
by Rovella to prove that there is a one-parameter family of maps whose
derivatives along their critical orbits increase exponentially fast and the
critical orbits have slow recurrent to the critical point. Metzger proved that
these maps have a unique absolutely continuous ergodic invariant probability
measure (SRB measure).
Here we use the technique developed by Freitas and show that the tail set
(the set of points which at a given time have not achieved either the
exponential growth of derivative or the slow recurrence) decays exponentially
fast as time passes. As a consequence, we obtain the continuous variation of
the densities of the SRB measures and associated metric entropies with the
parameter. Our main result also implies some statistical properties for these
maps.Comment: 1 figur
Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems
In this paper, we apply Devroye inequality to study various statistical
estimators and fluctuations of observables for processes. Most of these
observables are suggested by dynamical systems. These applications concern the
co-variance function, the integrated periodogram, the correlation dimension,
the kernel density estimator, the speed of convergence of empirical measure,
the shadowing property and the almost-sure central limit theorem. We proved in
\cite{CCS} that Devroye inequality holds for a class of non-uniformly
hyperbolic dynamical systems introduced in \cite{young}. In the second appendix
we prove that, if the decay of correlations holds with a common rate for all
pairs of functions, then it holds uniformly in the function spaces. In the last
appendix we prove that for the subclass of one-dimensional systems studied in
\cite{young} the density of the absolutely continuous invariant measure belongs
to a Besov space.Comment: 33 pages; companion of the paper math.DS/0412166; corrected version;
to appear in Nonlinearit
Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes
In this paper we consider horseshoes containing an orbit of homoclinic
tangency accumulated by periodic points. We prove a version of the Invariant
Manifolds Theorem, construct finite Markov partitions and use them to prove the
existence and uniqueness of equilibrium states associated to H\"older
continuous potentials.Comment: 33 pages, 6 figure
Slip on three-dimensional surfactant-contaminated superhydrophobic gratings
Trace amounts of surfactants have been shown to critically prevent the drag
reduction of superhydrophobic surfaces (SHSs), yet predictive models including
their effects in realistic geometries are still lacking. We derive theoretical
predictions for the velocity and resulting slip of a laminar fluid flow over
three-dimensional SHS gratings contaminated with surfactant, which allow for
the first direct comparison with experiments. The results are in good agreement
with our numerical simulations and with measurements of the slip in
microfluidic channels lined with SHSs, which we obtain via confocal microscopy
and micro-particle image velocimetry. Our model enables the estimation of a
priori unknown parameters of surfactants naturally present in applications,
highlighting its relevance for microfluidic technologies.Comment: 6 pages, 3 figures, 11 supplemental pages, 2 supplemental figure
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
Robust exponential decay of correlations for singular-flows
We construct open sets of Ck (k bigger or equal to 2) vector fields with
singularities that have robust exponential decay of correlations with respect
to the unique physical measure. In particular we prove that the geometric
Lorenz attractor has exponential decay of correlations with respect to the
unique physical measure.Comment: Final version accepted for publication with added corrections (not in
official published version) after O. Butterley pointed out to the authors
that the last estimate in the argument in Subsection 4.2.3 of the previous
version is not enough to guarantee the uniform non-integrability condition
claimed. We have modified the argument and present it here in the same
Subsection. 3 figures, 34 page
Statistical Properties and Decay of Correlations for Interval Maps with Critical Points and Singularities
We consider a class of piecewise smooth one-dimensional maps with critical
points and singularities (possibly with infinite derivative). Under mild
summability conditions on the growth of the derivative on critical orbits, we
prove the central limit theorem and a vector-valued almost sure invariance
principle. We also obtain results on decay of correlations.Comment: 18 pages, minor revisions, to appear in Communications in
Mathematical Physic
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