Shades of hyperbolicity for Hamiltonians

We prove that a Hamiltonian system H is an element of C-2(M, R) is globally hyperbolic if any of the following statements hold: 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 C-2-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 C-2-Hamiltonian is partially hyperbolic. Finally, we prove that stable weakly-shadowable regular energy hypersurfaces are partially hyperbolic.