Periodic points and rotation numbers for area preserving diffeomorphisms of the plane

Let f be an orientation preserving diffeomorphism of R ² which preserves area. We prove the existence of infinitely many periodic points with distinct rotation numbers around a fixed point of f, provided only that f has two fixed points whose infinitesimal rotation numbers are not both 0. We also show that if a fixed point z of f is enclosed in a "simple heteroclinic cycle" and has a non-zero infinitesimal rotation number r, then for every non-zero rational number p/q in an interval with endpoints 0 and r, there is a periodic orbit inside the heteroclinic cycle with rotation number p/q around z.