Shades of Hyperbolicity for Hamiltonians

TL;DR

This paper establishes conditions under which Hamiltonian systems are globally hyperbolic and explores the generic structure of energy hypersurfaces, highlighting the prevalence of partial hyperbolicity in robust dynamical behaviors.

Contribution

It proves that various stability and shadowing properties imply hyperbolicity in Hamiltonian systems and shows that, generically, energy hypersurfaces are densely composed of partially hyperbolic and elliptic orbits.

Findings

Robust topological stability implies global hyperbolicity.

Stably shadowable Hamiltonians are globally hyperbolic.

Partially hyperbolic structures are dense in generic Hamiltonian energy hypersurfaces.

Abstract

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.