ON THE GENERIC EXISTENCE OF PERIODIC ORBITS IN HAMILTONIAN DYNAMICS

We prove several generic existence results for infinitely many periodic orbits of Hamiltonian diffeomorphisms or Reeb flows. For example, we show that a Hamiltonian diffeomorphism of a complex projective space or Grassmannian generically has infinitely many periodic orbits. We also consider symplectomorphisms of the two-torus with irrational flux. We show that a symplectomorphism necessarily has infinitely many periodic orbits if it has one and all periodic points are nondegenerate.