Shades of hyperbolicity for Hamiltonians (Q2857759)

This paper explores aspects of hyperbolicity for Hamiltonian systems \(H:M\to\mathbb{R}\) on a symplectic \(2n\)-dimensional compact and connected manifold \(M\) with \(n\geq 2\). \(H\) is taken to be of class \(C^2\) on \(M\) throughout the paper; the authors' proofs rely on perturbation techniques only available in the \(C^2\)-topology.NEWLINENEWLINE The main result is the following: If \((H,e,{\mathcal E}_{H,e})\) is a Hamiltonian system with energy level \(e\) and with \({\mathcal E}_{H,e}\) a connected component of \(H^{-1}(e)\), then \((H,{\mathcal E}_{H,e})\) is Anosov if any of the following conditions hold:NEWLINENEWLINE (1) \((H,e,{\mathcal E}_{H,e})\) is robustly topologically stable;NEWLINENEWLINE (2) \((H,e,{\mathcal E}_{H,e})\) is stably shadowable;NEWLINENEWLINE (3) \((H,e,{\mathcal E}_{H,e})\) is stably expansive;NEWLINENEWLINE (4) \((H,e,{\mathcal E}_{H,e})\) has the stable weak specification property.NEWLINENEWLINE The proof depends on an intermediate result: if there is a neighborhood \(V\) of \((H,e,{\mathcal E}_{H,e})\) such that for any \((\overline H,\overline e,{\mathcal E}_{\overline H,\overline e})\) in \(V\), the corresponding regular energy hypersurface \({\mathcal E}_{\overline H,\overline e}\) has all hyperbolic closed orbits, then \((H,e,{\mathcal E}_{H,e})\) is Anosov.NEWLINENEWLINE The authors also prove that for a generic Hamiltonian \(H\in C^2(M,\mathbb{R})\), the union of the partially hyperbolic regular energy hypersurfaces and the closed elliptic orbits forms a dense subset of \(M\).