  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.
