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Abstract: 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 thatsymplectic structure by the standard kinetic energy Hamiltonian, and weconsider a compact regular energy level $\Sigma {k}:=H^{-1}k$ of $H$. Suppose$\Sigma {k}$ is an Anosov energy level. We prove that if $n$ is odd, then ifthe Hamiltonian flow restricted to $\Sigma {k}$ is Anosov with $C^{1}$ weakbundles then the Hamiltonian structure $\Sigma {k}$ is stable if and only ifit is contact. If $n$ is even and in addition the flow is assumed to be1-2-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 thetwisted symplectic structure is not exact then the Hamiltonian structure$\Sigma {k}$ is never stable for all sufficiently large $k$.

Autor: Will J. Merry, Gabriel P. Paternain



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