## CEMAPRE Seminar

### Stability and Shades of Hyperbolicity for Hamiltonians

Joana Torres
Abstract: Joint work with Mário Bessa and Jorge Rocha. We consider the setting of Hamiltonian systems, defined on a $2d$-dimensional symplectic manifold $M$ ($d \geq 2$). We address the following results, that we proved recently: - A Hamiltonian star system is Anosov. As a consequence we obtain the proof of the stability conjecture for Hamiltonians. This generalizes the $4$-dimensional results in [1]. -A Hamiltonian system $H$ is Anosov 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, for a $C^2$-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 $M$. As a consequence, any robustly transitive regular energy hypersurface of a $C^2$-Hamiltonian is partially hyperbolic. Finally, stably weakly shadowable regular energy hypersurfaces are partially hyperbolic. [1] M. Bessa, C. Ferreira and J. Rocha, On the stability of the set of hyperbolic closed orbits of a Hamiltonian, Math. Proc. Cambridge Philos. Soc., 149 (2) (2010), 373-383. [2] M. Bessa, J. Rocha and M. J. Torres, Hyperbolicity and Stability for Hamiltonian flows, Jr. Diff. Eq., 254 (1) (2013), 309-322. [3] M. Bessa, J. Rocha and M. J. Torres, Shades of Hyperbolicity for Hamiltonians, in preparation.