The Conley Conjecture

We prove the Conley conjecture for a closed symplectically aspherical symplectic manifold: a Hamiltonian diffeomorphism of a such a manifold has infinitely many periodic points. More precisely, we show that a Hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period. This theorem generalizes, for instance, a recent result of Hingston establishing the Conley conjecture for tori.