Shades of hyperbolicity for Hamiltonians

Abstract

Abstract. We prove that a Hamiltonian system H ∈ C 2 (M, R) 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 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 stably weakly shadowable regular energy hypersurfaces are partially hyperbolic.

On shadowing and hyperbolicity for geodesic flows on surfaces

“…Similar results are obtained when considering even feeble properties like the weak shadowing and the specification properties. Despite the Hamiltonian nature of the geodesic flow, the arguments in the present paper differ completely from those used in [5] for Hamiltonian systems.…”

“…The case of Hamiltonian systems yields some surprising consequences arising from numerical properties. If we assume C 2 -robustness of shadowing, then the closure of the periodic points is a uniformly hyperbolic set [5]. Thus, due to the general density theorem for Hamiltonians [20], the Hamiltonian is Anosov.…”

“…As a consequence we are not able to assure global hyperbolicity. Other techniques not available in the geodesic flow context are the so-called pasting lemma and suspension theorem which were crucial in [5].…”

Conservative flows with various types of shadowing

“…Another result which relates C 1 -robust properties with hyperbolicity is the result in [11] which states that C 1 -robustly transitive incompressible flows have dominated splitting. See also the results in [13,14,47] for flows and in [38,40] for diffeomorphisms.…”

Continuum-wise expansive symplectic diffeomorphisms

“…Bessa et al [8] shoed that if a Hamiltonian system is C 2 -robustly expansive then it is Anosov. In [7], Bessa and Rocha showed that a symplectic diffeomorphism is C 1 -robustly expansive then it is Anosov.…”