Nonuniform exponential dichotomies and Fredholm operators for flows

For the flow determined by a nonautonomous linear differential equation, we characterize the existence of a strong nonuniform exponential dichotomy in terms of the Fredholm property of a certain linear operator. We consider both cases of one-sided and two-sided exponential dichotomies. Moreover, we use the characterizations to establish the robustness of the notion of a strong nonuniform exponential dichotomy in a simple manner.     

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.     

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.     

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.     

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.     

Robust nonuniform dichotomies and parameter dependence

We establish the robustness of linear cocycles with an exponential dichotomy, under sufficiently small Lipschitz perturbations, in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations. We consider cocycles in Banach spaces, as well as the general case of nonuniform exponential dichotomies, and also the general case of an exponential behavior ecρ(n), given by an arbitrary sequence ρ(n) including the usual exponential behavior ρ(n)=n as a very special case. Moreover, we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter, and in fact that they are locally Lipschitz on finite-dimensional parameters.     

On nonuniform exponential dichotomy of evolution operators in Banach spaces

The problem of nonuniform exponential dichotomy of evolution operators in Banach spaces is considered. Connections between this concept and admissibility of the pair (C ₀,C ₀) are established. Generalizations to the nonuniform case of some results of Van Min, Räbiger and Schnaubelt ([MRS]) are obtained. It is shown that an implication from the uniform case is not true for nonuniform exponential dichotomy. The results are applicable for general time-varying linear equations with unbounded coefficients in Banach spaces.     

NOTES ON GENERALIZED EXPONENTIAL DICHOTOMIES

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Nonuniform exponential dichotomies and Lyapunov functions

For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.     

Comparative exponential dichotomies and column diagonal dominance

It is shown that the dichotomy obtained by Lazer for diagonally dominant linear systems is a weaker condition than that of exponential dichotomy but that the two conditions are equivalent in the case of bounded coefficients. Thus Fink's result that exponential decay of all solutions implies exponential dichotomy does not extend to the case of mixed growth and decay. It is also shown that column dominant systems of mixed sign admit exponential dichotomies when the coefficients are bounded.     

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.     

Lyapunov sequences for exponential dichotomies

For a linear nonautonomous dynamics with discrete time, we study the relation between nonuniform exponential dichotomies and strict Lyapunov sequences. Given such a sequence, we obtain the stable and unstable subspaces from the intersection of the images and preimages of the cones defined by each element of the sequence. The main difficulty is to extract some information about the angles between the stable and unstable subspaces (or some appropriate notion in the case of Banach spaces) from the Lyapunov sequence. In particular, for a large class of nonuniform exponential dichotomies we give a complete characterization in terms of strict quadratic Lyapunov sequences, that is, strict Lyapunov sequences defined by quadratic forms. We also construct explicitly families of strict Lyapunov sequences for each nonuniform exponential dichotomy, in terms of Lyapunov norms.     

Fredholm composition operators

Fredholm composition operators on a variety of Hilbert spaces of analytic functions on domains in , are characterized.

     

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.     

Fredholm weighted composition operators

We characterize the Fredholm weighted composition operators onC(X). In particular, ifX is a set with some regular property like intervals or balls inR ⁿ , our characterization implies that a weighted composition operator is Fredholm if and only if it is invertible. This equivalence is true for weighted composition operators onL ^p (), where is a nonatomic measure (1p<).     

Smooth robustness of parameterized perturbations of exponential dichotomies

We establish the robustness of nonuniform exponential dichotomies in Banach spaces, under sufficiently small C¹-parameterized perturbations. Moreover, we show that the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class C¹ on the parameter, thus yielding an optimal smoothness.