72 research outputs found
A Borel-Cantelli lemma for intermittent interval maps
We consider intermittent maps T of the interval, with an absolutely
continuous invariant probability measure \mu. Kim showed that there exists a
sequence of intervals A_n such that \sum \mu(A_n)=\infty, but \{A_n\} does not
satisfy the dynamical Borel-Cantelli lemma, i.e., for almost every x, the set
\{n : T^n(x)\in A_n\} is finite. If \sum \Leb(A_n)=\infty, we prove that
\{A_n\} satisfies the Borel-Cantelli lemma. Our results apply in particular to
some maps T whose correlations are not summable.Comment: 7 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
Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems
We consider a simple class of fast-slow partially hyperbolic dynamical
systems and show that the (properly rescaled) behaviour of the slow variable is
very close to a Friedlin--Wentzell type random system for times that are rather
long, but much shorter than the metastability scale. Also, we show the
possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon
that turns out to be related to the lack of absolutely continuity of the
central foliation.Comment: To appear in Journal of Statistical Physic
Operator renewal theory and mixing rates for dynamical systems with infinite measure
We develop a theory of operator renewal sequences in the context of infinite
ergodic theory. For large classes of dynamical systems preserving an infinite
measure, we determine the asymptotic behaviour of iterates of the
transfer operator. This was previously an intractable problem.
Examples of systems covered by our results include (i) parabolic rational
maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly
expanding interval maps with indifferent fixed points.
In addition, we give a particularly simple proof of pointwise dual ergodicity
(asymptotic behaviour of ) for the class of systems under
consideration.
In certain situations, including Pomeau-Manneville intermittency maps, we
obtain higher order expansions for and rates of mixing. Also, we obtain
error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a
minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated
version includes minor corrections in Sections 10 and 11, and corresponding
modifications of certain statements in Section 1. All main results are
unaffected. In particular, Sections 2-9 are unchanged from the published
versio
A strong pair correlation bound implies the CLT for Sinai Billiards
For Dynamical Systems, a strong bound on multiple correlations implies the
Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is
derived for dynamically Holder continuous observables of dispersing Billiards.
Here we weaken the regularity assumption and subsequently show that the bound
on multiple correlations follows directly from the bound on pair correlations.
Thus, a strong bound on pair correlations alone implies the CLT, for a wider
class of observables. The result is extended to Anosov diffeomorphisms in any
dimension.Comment: 13 page
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
Multiscale Systems, Homogenization, and Rough Paths:VAR75 2016: Probability and Analysis in Interacting Physical Systems
In recent years, substantial progress was made towards understanding
convergence of fast-slow deterministic systems to stochastic differential
equations. In contrast to more classical approaches, the assumptions on the
fast flow are very mild. We survey the origins of this theory and then revisit
and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1
(2016), 479-520], taking into account recent progress in -variation and
c\`adl\`ag rough path theory.Comment: 27 pages. Minor corrections. To appear in Proceedings of the
Conference in Honor of the 75th Birthday of S.R.S. Varadha
Absolutely continuous invariant measures for random non-uniformly expanding maps
We prove existence of (at most denumerable many) absolutely continuous
invariant probability measures for random one-dimensional dynamical systems
with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is
bounded away from zero, we obtain finitely many ergodic absolutely continuous
invariant probability measures, describing the asymptotics of almost every
point.
We also prove a similar result for higher-dimensional random non-uniformly
expanding dynamical systems. The results are consequences of the construction
of such measures for skew-products with essentially arbitrary base dynamics and
asymptotic expansion along the fibers. In both cases our method deals with
either critical or singular points for the random maps.Comment: 30 pages; 2 figures. Keywords: non-uniform expansion, random
dynamics, slow recurrence, singular and critical set, absolutely continuous
invariant measures, skew-product. To appear in Math Z, 201
Resonances in a chaotic attractor crisis of the Lorenz Flow
Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems. However, it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle. This behavior is investigated here for a boundary crisis in the Lorenz flow, for which neither the Lyapunov exponents nor the covariant Lyapunov vectors provide a criterion for the crisis. Instead, the convergence of the time evolution of probability densities to the invariant measure, governed by the semigroup of transfer operators, is expected to slow down at the approach of the crisis. Such convergence is described by the eigenvalues of the generator of this semigroup, which can be divided into two families, referred to as the stable and unstable Ruelle--Pollicott resonances, respectively. The former describes the convergence of densities to the attractor (or escape from a repeller) and is estimated from many short time series sampling the state space. The latter is responsible for the decay of correlations, or mixing, and can be estimated from a long times series, invoking ergodicity. It is found numerically for the Lorenz flow that the stable resonances do approach the imaginary axis during the crisis, as is indicative of the loss of global stability of the attractor. On the other hand, the unstable resonances, and a fortiori the decay of correlations, do not flag the proximity of the crisis, thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises
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