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We prove that any Hamiltonian diffeomorphism of a closed symplectic manifold equipped with an atoroidal symplectic form has simple non‐contractible periodic orbits of arbitrarily large period, provided that the diffeomorphism has a non‐degenerate (or even isolated and homologically non‐trivial) periodic orbit with non‐zero homology class and the set of one‐periodic orbits in that class is finite.

On non‐contractible periodic orbits of Hamiltonian diffeomorphisms