25 research outputs found
Partial holomorphic connections and extension of foliations
This paper stresses the strong link between the existence of partial
holomorphic connections on the normal bundle of a foliation seen as a quotient
of the ambient tangent bundle and the extendability of a foliation to an
infinitesimal neighborhood of a submanifold. We find the obstructions to
extendability and thanks to the theory developed we obtain some new
Khanedani-Lehmann-Suwa type index theorems
ON THE CONTINUITY OF LYAPUNOV EXPONENTS FOR SYSTEMS WITH ADDITIVE NOISE (Research on the Theory of Random Dynamical Systems and Fractal Geometry)
In this short essay I show a small result on continuity of Lyapunov exponents for systems with additive noise
How does noise induce order?
In this paper we present a general result with an easily check-able condition
that ensures a transition from chaotic regime to regular regimein random
dynamical systems with additive noise. We show how this resultapplies to a
prototypical family of nonuniformly expanding one dimensionaldynamical systems,
showing the main mathematical phenomenon behind Noise Induced Order.Comment: 17 pages, 3 figure
An elementary way to rigorously estimate convergence to equilibrium and escape rates
We show an elementary method to have (finite time and asymptotic) computer
assisted explicit upper bounds on convergence to equilibrium (decay of
correlations) and escape rate for systems satisfying a Lasota Yorke inequality.
The bounds are deduced by the ones of suitable approximations of the system's
transfer operator. We also present some rigorous experiment showing the
approach and some concrete result.Comment: 14 pages, 6 figure
Noise induced order for skew-products over a non-uniformly expanding base
Noise-induced order is the phenomenon by which the chaotic regime of a
deterministic system is destroyed in the presence of noise. In this manuscript,
we establish noise-induced order for a natural class of systems of dimension
consisting of a fiber-contracting skew product a over
nonuniformly-expanding 1-dimensional system.Comment: 17 pages, 2 figure
A Rigorous Computational Approach to Linear Response
We present a general setting in which the formula describing the linear
response of the physical measure of a perturbed system can be obtained. In this
general setting we obtain an algorithm to rigorously compute the linear
response. We apply our results to expanding circle maps. In particular, we
present examples where we compute, up to a pre-specified error in the
-norm, the response of expanding circle maps under stochastic and
deterministic perturbations. Moreover, we present an example where we compute,
up to a pre-specified error in the -norm, the response of the intermittent
family at the boundary; i.e., when the unperturbed system is the doubling map.Comment: Revised version following reports. A new example which contains the
computation of the linear response at the boundary of the intermittent family
has been adde
Rigorous approximation of diffusion coefficients for expanding maps
We use Ulam's method to provide rigorous approximation of diffusion
coefficients for uniformly expanding maps. An algorithm is provided and its
implementation is illustrated using Lanford's map.Comment: In this version Lanford's map has been used to illustrate the
computer implementation of the algorithm. To appear in Journal of Statistical
Physic
A general approach to Lehmann-Suwa-Khanedani index theorems: partial holomorphic connections and extensions of foliations
This thesis stresses the strong link between the existence of partial holomorphic connections on the normal bundle of a foliation seen as a quotient of the ambient tangent bundle and the extendability of a foliation to an infinitesimal neighborhood of a submanifold. We find some obstructions to extendability and thanks to the theory developed we obtain some new Khanedani-Lehmann-Suwa type index theorems, for foliations and holomorphic self maps
A general framework for the rigorous computation of invariant densities and the coarse-fine strategy
In this paper we present a general, axiomatical framework for the rigorous
approximation of invariant densities and other important statistical features
of dynamics. We approximate the system trough a finite element reduction, by
composing the associated transfer operator with a suitable finite dimensional
projection (a discretization scheme) as in the well-known Ulam method.
We introduce a general framework based on a list of properties (of the system
and of the projection) that need to be verified so that we can take advantage
of a so-called ``coarse-fine'' strategy. This strategy is a novel method in
which we exploit information coming from a coarser approximation of the system
to get useful information on a finer approximation, speeding up the
computation. This coarse-fine strategy allows a precise estimation of invariant
densities and also allows to estimate rigorously the speed of mixing of the
system by the speed of mixing of a coarse approximation of it, which can easily
be estimated by the computer.
The estimates obtained here are rigourous, i.e., they come with exact error
bounds that are guaranteed to hold and take into account both the
discretiazation and the approximations induced by finite-precision arithmetic.
We apply this framework to several discretization schemes and examples of
invariant density computation from previous works, obtaining a remarkable
reduction in computation time.
We have implemented the numerical methods described here in the Julia
programming language, and released our implementation publicly as a Julia
package
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