  We prove that for a certain class of closed monotone symplectic manifolds any
Hamiltonian diffeomorphism with a hyperbolic fixed point must necessarily have
infinitely many periodic orbits. Among the manifolds in this class are complex
projective spaces, some Grassmannians, and also certain product manifolds such
as the product of a projective space with a symplectically aspherical manifold
of low dimension.
  A key to the proof of this theorem is the fact that the energy required for a
Floer connecting trajectory to approach an iterated hyperbolic orbit and cross
its fixed neighborhood is bounded away from zero by a constant independent of
the order of iteration. This result, combined with certain properties of the
quantum product specific to the above class of manifolds, implies the existence
of infinitely many periodic orbits.
