Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in R-4

In this article we study Hamiltonian flows associated to smooth functions H : R-4 -> R restricted to energy levels close to critical levels. We assume the existence of a saddle-center equilibrium point p(c) in the zero energy level H-1( 0). The Hamiltonian function near p(c) is assumed to satisfy Moser's normal form and pc is assumed to lie in a strictly convex singular subset S-0 of H-1(0). Then for all E > 0 small, the energy level H-1(E) contains a subset S-E near S-0, diffeomorphic to the closed 3-ball, which admits a system of transversal sections F-E, called a 2 - 3 foliation. F-E is a singular foliation of SE and contains two periodic orbits P-2,P-E subset of partial derivative S-E and P-3,P-E subset of S-E \textbackslash{} partial derivative S-E as binding orbits. P-2,P-E is the Lyapunoff orbit lying in the center manifold of p(c), has Conley-Zehnder index 2 and spans two rigid planes in partial derivative S-E. P-3,P-E has Conley-Zehnder index 3 and spans a one parameter family of planes in S-E \textbackslash{} partial derivative S-E. A rigid cylinder connecting P-3,P-E to P-2,P-E completes F-E. All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to P-2,P-E in S-E \textbackslash{} partial derivative S-E follows from this foliation. (AU)