Resumen de Integral stable manifolds in Banach spaces

Luis M. Barreira, C. Silva, Clàudia Valls Anglès

• We establish the existence of smooth integral stable manifolds for sufficiently small perturbations of nonuniform exponential dichotomies in Banach spaces. We also consider the case of a nonautonomous dynamics given by a sequence of C1 maps. The optimal smoothness of the manifolds is obtained at the same time as their existence, using a convenient lemma of Henry. Furthermore, we obtain not only the exponential decay of the dynamics along the stable manifolds, but also of its derivative. In addition, we give a characterization of the stable manifolds in terms of the maximal exponential growth rate that is allowed, we discuss how the manifolds vary with the perturbations, and we discuss their equivariance with respect to a sequence of linear operators.