In this paper, we prove certain persistence properties of
the homoclinic
points in Hamiltonian systems and symplectic
diffeomorphisms. We show that, under
some general conditions, stable and unstable manifolds
of hyperbolic periodic points
intersect in a very persistent way and we also give
some simple criteria for positive topological entropy.
The method used is the intersection theory of Lagrangian
submanifolds of symplectic manifolds.