TITLE:
Contribution to the study of invariant manifolds and
the splitting of separatrices of parabolic points.

AUTHORS:
I. Baldoma

Departament de Matematica Aplicada i Analisi.
Universitat de Barcelona.
Gran Via 585, 08007, Barcelona, Spain.
E-Mail: barraca@cerber.mat.ub.es

ABSTRACT:

This thesis has been directed by Prof. Ernest Fontich.

We study two problems: the splitting of separatrices and
the existence and regularity of invariant manifolds associated
to parabolic fixed points.

Concerning the first problem, we consider a class of rapidly forced
Hamiltonian systems with one and a half degrees of freedom having a
fixed point with a double zero eigenvalue but not diagonalizable.
We assume that for some value of the parameter the system is autonomous
and has a homoclinic connexion associated to the fixed point.
We prove an asymptotic formula to measure the splitting of separatrices
which is exponentially small with respect to the frequency of
the perturbation.

Concerning the second problem, we give sufficient conditions for
the existence of a stable invariant manifold for a mapping in a
n-dimensional space with a fixed point such that the derivative of the
mapping is the identity. We consider the Lipschitz and the analytical
cases and we prove that the stable invariant manifold is Lipschitz
and analytical respectively in some suitable domains.