We prove several generic existence results for infinitely many periodic
orbits of Hamiltonian diffeomorphisms or Reeb flows. For instance, 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 such a
symplectomorphism necessarily has infinitely many periodic orbits whenever it
has one and all periodic points are non-degenerate.