Generic Dynamics of 4-Dimensional C 2 Hamiltonian Systems

We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C ²-residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each either being Anosov or having zero Lyapunov exponents almost everywhere. This is in the spirit of the Bochi-Mañé dichotomy for area-preserving diffeomorphisms on compact surfaces 2] and its continuous-time version for 3-dimensional volume-preserving flows 1].