  We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic
manifold with semisimple quantum homology has more contractible fixed points,
counted homologically, than the total dimension of the homology of the
manifold, then it must have an infinite number of contractible periodic points.
This constitutes a higher-dimensional homological generalization of a
celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in
1994.
