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For any finite number of points, the subgroup of Hamiltonian symplectomorphism that fixes some neighbourhood of the points acts transitively on the complement of the points.

Choose two points $a, b$ in the complement, and a path avoiding the constrained points joining them. Then you can find a (non-autonomous) Hamiltonian function with support in an arbitrarily small neighborhood of the path whose time-1 flow maps $a$ to $b$. Indeed, the group of Hamiltonian tranformations with support in a connected neighbourhood is transitive on Darboux balls charts, so the orbit of a point is open and closed.

What this argument seems to prove is in fact the statement :

For any connected open subset $\Omega$ in a symplectic manifold, the group of Hamiltonian symplectomorphisms with support in $\Omega$ is transitive in $\Omega$.

In particular, the group of compactly Hamiltonian symplectomorphisms is $n$-transitive $\forall n$ on any connected symplectic manifold.

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For any finite number of points, the subgroup of Hamiltonian symplectomorphism that fixes some neighbourhood of the points acts transitively on the complement of the points.

Choose two points $a, b$ in the complement, and a path avoiding the constrained points joining them. Then you can find a (non-autonomous) Hamiltonian function with support in an arbitrarily small neighborhood of the path whose time-1 flow maps $a$ to $b$. Indeed, the group of Hamiltonian tranformations with support in a connected neighbourhood is transitive on Darboux balls charts, so the orbit of a point is open and closed.