The main result of this paper is that every non-trivial Hamiltonian diffeomorphism of a closed oriented surface of genus at least one has periodic points of arbitrarily high period. The same result is true for S2 provided the diffeomorphism has at least three fixed points. In addition we show that up to isotopy relative to its fixed point set, every orientation preserving diffeomorphism F : S →...

We focus on a very specific case of the Uniform Boundedness Conjecture, namely, bounding the number of possible c such that the quadratic polynomial φc(z) = z2 + c has quadratic periodic points of some small period. We show that there are infinitely many rational c with quadratic 4-cycles, with all such c completely understood; and only finitely many rational c with quadratic 5-cycles (we conje...

The study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifoldM , then it is classically known that there are at least dim M 2 +1 fixed points; this follows fromMorse theory for the momentum map of the action. In this paper we use Atiyah-Bott-Berline-Vergne (ABBV) localization in equivariant cohomology to p...