Hamiltonian suspension of perturbed Poincare sections and an application

We construct a Hamiltonian suspension for a given symplectomorphism which is the perturbation of a Poincare map. This is especially useful for the conversion of perturbative results between symplectomorphisms and Hamiltonian flows in any dimension 2d. As an application, using known properties of area-preserving maps, we prove that for any Hamiltonian defined on a symplectic 4-manifold M and any point p is an element of M, there exists a C-2-close Hamiltonian whose regular energy surface through p is either Anosov or contains a homo-clinic tangency.