# Stability of Anosov Hamiltonian Structures - Mathematics > Dynamical Systems

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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

Fuente: https://arxiv.org/