  We study Hamiltonian diffeomorphisms of closed symplectic manifolds with
non-contractible periodic orbits. In a variety of settings, we show that the
presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism
of a closed toroidally monotone or toroidally negative monotone symplectic
manifold implies the existence of infinitely many non-contractible periodic
orbits in a specific collection of free homotopy classes. The main new
ingredient in the proofs of these results is a filtration of Floer homology by
the 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 action
filtration for atoroidal manifolds.
