= Hamiltonian transitivity on a connected symplectic manifold
Compactly supported <Hamiltonian vector fields> can attain any prescribed tangent vector at a point of a <symplectic manifold>: prescribe the derivative of a local Hamiltonian and multiply by a <bump function>. Composing flows for a tangent <basis> gives an open orbit by the <inverse function theorem>. Every orbit is open, so on a <connected> manifold the compactly supported Hamiltonian transformations act transitively.
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