Evolution maps and center manifolds
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Aequationes Mathematicae
Evolution maps and center manifolds Luis Barreira
and Claudia Valls
Abstract. We use evolution maps to construct center-stable, center-unstable and center invariant manifolds with optimal regularity for a nonautonomous dynamics with discrete time. The proofs comprise essentially two steps: first we reformulate the original problem in terms of an autonomous problem involving evolutions maps, and then we apply an autonomous result to construct autonomous invariant manifolds that we are able to push back to the original problem. We also describe how the invariant manifolds vary with the perturbations. The main novelty of our work is the method of proof, which leads to simple proofs. Mathematics Subject Classification. Primary 34D09, 37D25, 47D06. Keywords. Evolution maps, Families of norms, Nonuniform hyperbolicity.
1. Introduction 1.1. Evolution maps The notion of hyperbolicity plays a central role in a large part of the stability theory of differential equations and dynamical systems. The same holds for its many variants, such as partial hyperbolicity, nonuniform hyperbolicity and general growth rates. The presence of a certain amount of hyperbolicity allows one to construct for example topological conjugacies and stable and unstable invariant manifolds, under sufficiently small nonlinear perturbations and mild additional assumptions. It is thus quite relevant to characterize hyperbolicity in various ways that in different situations may be more amenable or useful than others. Here we consider a description of hyperbolicity in terms of evolution maps (following a corresponding approach for continuous time using evolution semigroups, as described for example in the books [6,12]). The idea of a similar approach goes Partially supported by FCT/Portugal through UID/MAT/04459/2019.
L. Barreira, C. Valls
AEM
back to Mather [10]. A main advantage of this approach is that it can be used when the involved linear operators are unbounded and when the dynamics is nonautonomous. More precisely, let A be a bounded linear operator on a Banach space X. Then the exponential behavior of the dynamics xn+1 = Axn
for n ∈ Z
can be characterized in terms of the spectrum σ(A) of A. On the other hand, for a nonautonomous dynamics xn+1 = An xn
for n ∈ Z
(1)
defined by a sequence of linear operators An we need to look for alternative characterizations. One possibility is to consider the evolution map T defined by (T x)n = An−1 xn−1
for n ∈ Z,
with x = (xn )n∈Z in some appropriate Banach space of sequences in X. It turns out that the hyperbolicity of the original dynamics in (1) can be characterized in terms of the hyperbolicity of the operator T (see [2]). Note that the evolution map T defines an autonomous dynamics, while the original dynamics in (1) may be nonautonomous. It goes without saying that it is in general simpler to deal with an autonomous dynamics. This implies that it is often quite convenient to deal first with a given property for the evolution map T and then (try to) transfer this property to th
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