112 research outputs found
Multifractals via recurrence times ?
This letter is a comment on an article by T.C. Halsey and M.H. Jensen in
Nature about using recurrence times as a reliable tool to estimate multifractal
dimensions of strange attractors. Our aim is to emphasize that in the recent
mathematical literature (not cited by these authors), there are positive as
well as negative results about the use of such techniques. Thus one may be
careful in using this tool in practical situations (experimental data).Comment: This is a very short and non-technical note written after an article
published in Nature by T.C. Halsey and M.H. Jense
Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors
We consider two dimensional maps preserving a foliation which is uniformly
contracting and a one dimensional associated quotient map having exponential
convergence to equilibrium (iterates of Lebesgue measure converge exponentially
fast to physical measure). We prove that these maps have exponential decay of
correlations over a large class of observables. We use this result to deduce
exponential decay of correlations for the Poincare maps of a large class of
singular hyperbolic flows. From this we deduce logarithm laws for these flows.Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos
corrected; improvements on the statements and comments suggested by a
referee. Keywords: singular flows, singular-hyperbolic attractor, exponential
decay of correlations, exact dimensionality, logarithm la
An elementary approach to rigorous approximation of invariant measures
We describe a framework in which is possible to develop and implement
algorithms for the approximation of invariant measures of dynamical systems
with a given bound on the error of the approximation.
Our approach is based on a general statement on the approximation of fixed
points for operators between normed vector spaces, allowing an explicit
estimation of the error.
We show the flexibility of our approach by applying it to piecewise expanding
maps and to maps with indifferent fixed points. We show how the required
estimations can be implemented to compute invariant densities up to a given
error in the or distance. We also show how to use this to
compute an estimation with certified error for the entropy of those systems.
We show how several related computational and numerical issues can be solved
to obtain working implementations, and experimental results on some one
dimensional maps.Comment: 27 pages, 10 figures. Main changes: added a new section in which we
apply our method to Manneville-Pomeau map
Arnold maps with noise: Differentiability and non-monotonicity of the rotation number
Arnold's standard circle maps are widely used to study the quasi-periodic
route to chaos and other phenomena associated with nonlinear dynamics in the
presence of two rationally unrelated periodicities. In particular, the El
Nino-Southern Oscillation (ENSO) phenomenon is a crucial component of climate
variability on interannual time scales and it is dominated by the seasonal
cycle, on the one hand, and an intrinsic oscillatory instability with a period
of a few years, on the other. The role of meteorological phenomena on much
shorter time scales, such as westerly wind bursts, has also been recognized and
modeled as additive noise. We consider herein Arnold maps with additive,
uniformly distributed noise. When the map's nonlinear term, scaled by the
parameter , is sufficiently small, i.e. , the map is
known to be a diffeomorphism and the rotation number is a
differentiable function of the driving frequency . We concentrate on
the rotation number's behavior as the nonlinearity becomes large, and show
rigorously that is a differentiable function of ,
even for , at every point at which the noise-perturbed map is
mixing. We also provide a formula for the derivative of the rotation number.
The reasoning relies on linear-response theory and a computer-aided proof. In
the diffeomorphism case of , the rotation number
behaves monotonically with respect to . We show, using again a
computer-aided proof, that this is not the case when and the
map is not a diffeomorphism.Comment: Electronic copy of final peer-reviewed manuscript accepted for
publication in the Journal of Statistical Physic
Recurrence and algorithmic information
In this paper we initiate a somewhat detailed investigation of the
relationships between quantitative recurrence indicators and algorithmic
complexity of orbits in weakly chaotic dynamical systems. We mainly focus on
examples.Comment: 26 pages, no figure
A "metric" complexity for weakly chaotic systems
We consider the number of Bowen sets which are necessary to cover a large
measure subset of the phase space. This introduce some complexity indicator
characterizing different kind of (weakly) chaotic dynamics. Since in many
systems its value is given by a sort of local entropy, this indicator is quite
simple to be calculated. We give some example of calculation in nontrivial
systems (interval exchanges, piecewise isometries e.g.) and a formula similar
to the Ruelle-Pesin one, relating the complexity indicator to some initial
condition sensitivity indicators playing the role of positive Lyapunov
exponents.Comment: 15 pages, no figures. Articl
Khinchin theorem for integral points on quadratic varieties
We prove an analogue the Khinchin theorem for the Diophantine approximation
by integer vectors lying on a quadratic variety. The proof is based on the
study of a dynamical system on a homogeneous space of the orthogonal group. We
show that in this system, generic trajectories visit a family of shrinking
subsets infinitely often.Comment: 19 page
Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables
This is the author accepted manuscript. The final version is available from the American Institute of Mathematical Sciences via the DOI in this recordWe establish quantitative results for the statistical behaviour of
infinite systems. We consider two kinds of infinite system:
i) a conservative dynamical system (f, X, µ) preserving a σ-finite measure µ
such that µ(X) = ∞;
ii) the case where µ is a probability measure but we consider the statistical behaviour of an observable φ: X → [0, ∞) which is non-integrable:
R
φ dµ = ∞.
In the first part of this work we study the behaviour of Birkhoff sums of
systems of the kind ii). For certain weakly chaotic systems, we show that these
sums can be strongly oscillating. However, if the system has superpolynomial
decay of correlations or has a Markov structure, then we show this oscillation
cannot happen. In this case we prove a general relation between the behavior of
φ, the local dimension of µ, and the scaling rate of the growth of Birkhoff sums
of φ as time tends to infinity. We then establish several important consequences
which apply to infinite systems of the kind i). This includes showing anomalous
scalings in extreme event limit laws, or entrance time statistics. We apply our
findings to non-uniformly hyperbolic systems preserving an infinite measure,
establishing anomalous scalings for the power law behavior of entrance times
(also known as logarithm laws), dynamical Borel–Cantelli lemmas, almost sure
growth rates of extremes, and dynamical run length functions.Engineering and Physical Sciences Research Council (EPSRC)GNAMPA-INdAMUniversity of PisaUniversity of HoustonInstitut Mittag-LefflerNSFCUniversity of WarwickUniversity of ExeterCentre Physique Theorique, Marseill
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