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Stability of Anosov Hamiltonian Structures

Consider the tangent bundle of a Riemannian manifold $(M,g)$ of dimension $n\geq3$ admitting a metric of negative curvature (not necessarily equal to $g$) endowed with a twisted symplectic structure defined by a closed 2-form on $M$. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by the standard kinetic energy Hamiltonian, and we consider a compact regular energy level $Σ_{k}:=H^{-1}(k)$ of $H$. Suppose $Σ_{k}$ is an Anosov energy level. We prove that if $n$ is odd, then if the Hamiltonian flow restricted to $Σ_{k}$ is Anosov with $C^{1}$ weak bundles then the Hamiltonian structure $(Σ_{k}$ is stable if and only if it is contact. If $n$ is even and in addition the flow is assumed to be 1/2-pinched then the same conclusion holds. As a corollary we deduce that if $g$ is negatively curved, strictly 1/4-pinched and the 2-form defining the twisted symplectic structure is not exact then the Hamiltonian structure $(Σ_{k}$ is never stable for all sufficiently large $k$.