A Grobman-Hartman theorem for general nonuniform exponential dichotomies

For a nonautonomous dynamics with discrete time given by a sequence of linear operators Am, we establish a version of the Grobman-Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(n), determined by a sequence ρ(n). For all sufficiently small Lipschitz perturbations Am+fm we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators Am. We also show that all conjugacies are Hölder continuous. We note that the usual exponential behavior is included as a very special case when ρ(n)=n, but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ(n)=logn.     

Robustness of general dichotomies

For a linear equation v^′=A(t)v we consider general dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates ecρ(t) for some function ρ(t). This includes as a special case the usual exponential behavior when ρ(t)=t. We also consider the general case of nonuniform exponential dichotomies. We establish the robustness of the exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a given linear equation persists under sufficiently small linear perturbations. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy.     

Stable manifolds for nonuniform polynomial dichotomies

We establish the existence of smooth stable manifolds in Banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy. This new dichotomy is more restrictive in the “nonuniform part” but allow the “uniform part” to obey a polynomial law instead of an exponential (more restrictive) law. We consider two families of perturbations. For one of the families we obtain local Lipschitz stable manifolds and for the other family, assuming more restrictive conditions on the perturbations and its derivatives, we obtain C¹ global stable manifolds. Finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family.     

Existence of nonuniform exponential dichotomies and a Fredholm alternative

We consider linear equations x^′=A(t)x in a Banach space, with nonuniform exponential dichotomies in and , and we give an admissibility condition for the existence of nonuniform exponential dichotomies in the whole line under sufficiently small perturbations. We also obtain a Fredholm alternative for the existence of bounded solutions under nonlinear perturbations.     

Robustness of nonuniform exponential trichotomies in Banach spaces

We establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x^′=A(t)x in Banach spaces. We also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy. We consider both trichotomies in semi-infinite intervals and trichotomies in R.     

Normal forms via nonuniform hyperbolicity

We develop a normal form theory for a nonautonomous dynamics $x_{m+1}=A_m{x}_m+f_m(x_m)$ with discrete time, based on the nonuniform spectrum of the sequence of matrices $A_m$. In particular, we show that any nonresonant terms of the perturbations $f_m$ can be eliminated through an appropriate coordinate change, with the resonances expressed in terms of the connected components of the nonuniform spectrum. The latter is defined in terms of the notion of a nonuniform exponential dichotomy with a small nonuniform part, which is ubiquitous in the context of ergodic theory. We first make a preparation of the linear part of the dynamics that is of independent interest: we show that any sequence of matrices with a bounded nonuniform spectrum can be reduced to a sequence of matrices in block form via a Lyapunov coordinate change. This allows maintaining the Lyapunov exponents as well as the nonuniform spectrum. As further developments, we describe normal form theories in two additional contexts: we consider nonuniformly hyperbolic cocycles over a diffeomorphism of a compact manifold as well as perturbations $f_m$ of a sequence of compact linear operators $A_m$ on a Banach space. The latter includes the particular case of a sequence of matrices that need not be invertible.     

Robust and bias-corrected estimation of the coefficient of tail dependence

We introduce a robust and asymptotically unbiased estimator for the coefficient of tail dependence in multivariate extreme value statistics. The estimator is obtained by fitting a second order model to the data by means of the minimum density power divergence criterion. The asymptotic properties of the estimator are investigated. The efficiency of our methodology is illustrated on a small simulation study and by a real dataset from the actuarial context.     

Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities

This paper is devoted to the study of nonlinear singular and non-singular fourth order difference equations coupled with periodic boundary value conditions. In the paper some existence and nonexistence results are given. The results are deduced from Krasnoselskii's fixed point theorems in cones. An exhaustive study of the Green's function related to a linear fourth order problem is done.     

Quadratic Lyapunov functions and nonuniform exponential dichotomies

For nonautonomous linear equations x^′=A(t)x, we show how to characterize completely nonuniform exponential dichotomies using quadratic Lyapunov functions. The characterization can be expressed in terms of inequalities between matrices. In particular, we obtain converse theorems, by constructing explicitly quadratic Lyapunov functions for each nonuniform exponential dichotomy. We note that the nonuniform exponential dichotomies include as a very special case (uniform) exponential dichotomies. In particular, we recover in a very simple manner a complete characterization of uniform exponential dichotomies in terms of quadratic Lyapunov functions. We emphasize that our approach is new even in the uniform case.Furthermore, we show that the instability of a nonuniform exponential dichotomy persists under sufficiently small perturbations. The proof uses quadratic Lyapunov functions, and in particular avoids the use of invariant unstable manifolds which, to the best of our knowledge, are not known to exist in this general setting.     

The Density of Linear Symplectic Cocycles with Simple Lyapunov Spectrum in YIC（X, SL（2, R））

Let （X, G（X）, m） be a probability space with a-algebra G（X） and probability measure m. The set V in G is called P-admissible, provided that for any positive integer n and positive-measure set Vn∈ contained in V, there exists a Zn∈G such that Zn belong to Vn and 0 〈 m（Zn） 〈 1/n. Let T be an ergodic automorphism of （X, G） preserving m, and A belong to the space of linear measurable symplectic cocycles     

Monotonicity on nonuniform grids

In this paper we study a class of numerical methods used to solve two-point boundary value problems on nonuniform grids. Particular attention is devoted to positive solutions, i.e. conditions under which the solutions of the problem are positive. Applications to steady states of air pollution problems are also referred to.     

Robustness of exponential dichotomies in mean

We consider the notion of an exponential dichotomy in mean, in which the exponential behavior in the classical notion of an exponential dichotomy is replaced by the much weaker requirement that the same happens in mean with respect to some probability measure. This includes as a special case any linear cocycle over a measure-preserving flow with nonzero Lyapunov exponents almost everywhere, such as the geodesic flow on a compact manifold of negative curvature. Our main aim is to show that the exponential behavior in mean is robust, in the sense that it persists under sufficiently small linear perturbations.     

Robustness of nonuniform exponential dichotomies in Banach spaces

We give conditions for the robustness of nonuniform exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a given linear equation x^′=A(t)x persists under a sufficiently small linear perturbation. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the “angles” between the stable and unstable subspaces. Our proofs exhibit (implicitly) the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions. We emphasize that we do not need the notion of admissibility (of bounded nonlinear perturbations). We also obtain related robustness results in the case of nonuniform exponential contractions. In addition, we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time.     

Center manifolds under nonuniform hyperbolicity - maximal regularity

We establish the existence of (invariant) center manifolds with maximal Cr regularity for a nonautonomous dynamics with discrete time. We consider the general case of perturbations of a nonuniform exponential trichotomy. Our proof uses the fiber contraction principle and allows linear perturbations without any further effort.     

Wavelet packets associated with nonuniform multiresolution analyses

A multiresolution analysis was defined by Gabardo and Nashed for which the translation set is a discrete set which is not a group. We construct the associated wavelet packets for such an MRA. Further, from the collection of dilations and translations of the wavelet packets, we characterize the subcollections which form orthonormal bases for L²(R).     

Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy

We establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that the linear equation v^′=A(t)v admits a nonuniform exponential dichotomy. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of Ck functions with α-Hölder continuous kth derivative. This is a closed subset of the space of continuous functions with the supremum norm, by an apparently not so well-known lemma of Henry (see Proposition 3). The estimates showing that the functions maintain the original bounds when transformed under the fixed-point operator are obtained through a careful application of the Faà di Bruno formula for the higher derivatives of the compositions (see (31) and (35)). As a consequence, we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class Ck.     

Wavelets associated with nonuniform multiresolution analyses and one-dimensional spectral pairs

A generalization of Mallat's classic multiresolution analysis (MRA), based on the theory of spectral pairs, was considered in two papers by Gabardo and Nashed. In this nonstandard setting, the translation set is no longer a subgroup or a translate of a subgroup of R, but is a spectrum associated with a one-dimensional spectral pair. In this paper, we continue the study based on this nonstandard setting and give the characterization for nonuniform wavelets associated with a nonuniform MRA. These characterizations are consistent with both the known necessary and sufficient conditions for the existence of nonuniform MRA wavelets and the known characterization for standard dyadic wavelets associated with an MRA.