  We prove that a C2 Hamiltonian system H in M is globally hyperbolic if any of
the following statements holds: H is robustly topologically stable; H is stably
shadowable; H is stably expansive; and H has the stable weak specification
property. Moreover, we prove that, for a C2-generic Hamiltonian H, the union of
the partially hyperbolic regular energy hypersurfaces and the closed elliptic
orbits, forms a dense subset of M. As a consequence, any robustly transitive
regular energy hypersurface of a C2-Hamiltonian is partially hyperbolic.
Finally, we prove that stably weakly shadowable regular energy hypersurfaces
are partially hyperbolic.
