Hamiltonian suspension of perturbed Poincaré sections and an application

Abstract

Abstract. We construct a Hamiltonian suspension for a given symplectomorphism which is the perturbation of a Poincaré 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 ∈ M , there exists a C 2 -close Hamiltonian whose regular energy surface through p is either Anosov or contains a homoclinic tangency.

On shadowing and hyperbolicity for geodesic flows on surfaces

“…The proof is contained in section 5. We point out that the proof of the Hamiltonian version of this theorem in [5] uses a suspension theorem [4] which is unavailable for geodesic flows. Notice also that the general Hamiltonian version of Theorem 3 contained in [5] is stronger because it requires the closing lemma, unknown for geodesic flows.…”