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

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 $L^{\infty}$-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 $L^1$-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

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 $\geq 2$ consisting of a fiber-contracting skew product a over nonuniformly-expanding 1-dimensional system.Comment: 17 pages, 2 figure

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 $L^{1}$ or $L^\infty$ 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