On the Stability of the Set of Hyperbolic Closed Orbits of a Hamiltonian

A Hamiltonian level, say a pair $(H,e)$ of a Hamiltonian $H$ and an energy $e \in $\mathbb{R}$$,issaidtobeAnosovifthereexistsaconnectedcomponent$$\mathcal{E}$_{H,e}$of$H^{-1}({e})$ which is uniformly hyperbolic for the Hamiltonian flow $X_H^t$. The pair $(H,e)$ is said to be a Hamiltonian star system if there exists a connected component ${\mathcal{E}}_{H,e}^{\star }$ of the energy level $H^{-1}(e)$ such that all the closed orbits and all the critical points of ${\mathcal{E}}_{H,e}^{\star }$ are hyperbolic, and the same holds for a connected component of the energy level ${\tilde{H}}^{-1}(\tilde{e})$, close to ${\mathcal{E}}_{H,e}^{\star }$, for any Hamiltonian $\tilde{H}$, in some $C^2$-neighbourhood of $H$, and $\tilde{e}$ in some neighbourhood of $e$. In this article we prove that for any four-dimensional Hamiltonian star level $(H,e)$ if the surface ${\mathcal{E}}_{H,e}^{\star }$ does not contain critical points, then $X_H^t{\mathrm{\mid }}_{{\mathcal{E}}_{H,e}^{\star }}$ is Anosov; if ${\mathcal{E}}_{H,e}^{\star }$ has critical points, then there exists $\tilde{e}$, arbitrarily close to $e$, such that $X_H^t{\mathrm{\mid }}_{{\mathcal{E}}_{H,\tilde{e}}^{\star }}$ is Anosov.