  We show that the actions and indexes of fixed points of a Hamiltonian
diffeomorphism with finitely many periodic points must satisfy certain
relations, provided that the quantum cohomology of the ambient manifold meets
an algebraic requirement satisfied for projective spaces, Grassmannians and
many other manifolds. We also refine a previous result on the Conley conjecture
for negative monotone symplectic manifolds, due to the second and third
authors, and show that a Hamiltonian diffeomorphism of such a manifold must
have simple periodic orbits of arbitrarily large period whenever its fixed
points are isolated.
