Conley Conjecture Revisited

We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree 2 with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic Calabi-Yau and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class). The proof hinges on a general Lusternik-Schnirelmann type result that, under some natural additional conditions, the sequence of mean spectral invariants for the iterations of a Hamiltonian diffeomorphism never stabilizes. We also show that for the iterations of a Hamiltonian diffeomorphism with finitely many periodic orbits the sequence of action gaps between the largest and the smallest spectral invariants remains bounded and, as a consequence, establish some new cases of the -generic existence of infinitely many simple periodic orbits.