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We prove that any perturbation of the symplectic part of the derivative of aPoisson diffeomorphism can be realized as the derivative of a $C^1$-closePoisson diffeomorphism. We also show that a similar property holds for thePoincar\'e map of a Hamiltonian on a Poisson manifold. These results are theconservative counterparts of the Franks lemma, a perturbation tool used inseveral contexts most notably in the theory of smooth dynamical systems.

Realization of tangent perturbations in discrete and continuous time conservative systems