Creation of Homoclinic Tangencies in Hamiltonians by the Suspension of Poincar\'e Sections

Abstract

In this note we show that for any Hamiltonian defined on a symplectic 4-manifold M and any point p in M, there exists a C2-close Hamiltonian whose regular energy surface through p is either Anosov or it contains a homoclinic tangency. Our result is based on a general construction of Hamiltonian suspensions for given symplectomorphisms on Poincar\'e sections already known to yield similar properties.