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We study Hamiltonian diffeomorphisms of closed symplectic manifolds withnon-contractible periodic orbits. In a variety of settings, we show that thepresence of one non-contractible periodic orbit of a Hamiltonian diffeomorphismof a closed toroidally monotone or toroidally negative monotone symplecticmanifold implies the existence of infinitely many non-contractible periodicorbits in a specific collection of free homotopy classes. The main newingredient in the proofs of these results is a filtration of Floer homology bythe so-called augmented action. This action is independent of capping, and,under favorable conditions, the augmented action filtration for toroidally(negative) monotone manifolds can play the same role as the ordinary actionfiltration for atoroidal manifolds.

Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds