  A celebrated theorem in two-dimensional dynamics due to John Franks asserts
that every area preserving homeomorphism of the sphere has either two or
infinitely many periodic points. In this work we reprove Franks' theorem under
the additional assumption that the map is smooth. Our proof uses only tools
from symplectic topology and thus differs significantly from all previous
proofs. A crucial role is played by the results of Ginzburg and Kerman
concerning resonance relations for Hamiltonian diffeomorpisms.
