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Journal of the London Mathematical Society Advance Access published online on February 20, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm120
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© 2007 London Mathematical Society

Integral stable manifolds in Banach spaces

L. Barreira and C. Silva

Departamento de Matemática
Instituto Superior Técnico
1049-001 Lisboa
Portugal
http://www.math.ist.utl.pt/~barreira/
ecvalls@math.ist.utl.pt

C. Valls

Departamento de Matemática
Universidade da Beira Interior
Rua Mar-quês d’Ávila e Bolama
6201-001 Covilhã
Portugal
csilva@mat.ubi.pt
http://www.mat.ubi.pt/~csilva/

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.

Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundação para a Cîencia e a Tecnologia by the Programs POCTI/FEDER, POSI and POCI 2010/Fundo Social Europeu, and the grant SFRH/BPD/26465/2006.

2000 Mathematics Subject Classification 37D10, 37D25 (primary).

Received December 18, 2006; revised September 27, 2007;
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