# American Institute of Mathematical Sciences

October  2009, 3(4): 595-610. doi: 10.3934/jmd.2009.3.595

## On the generic existence of periodic orbits in Hamiltonian dynamics

 1 Department of Mathematics, University of California at Santa Cruz, Santa Cruz, CA 95064, United States 2 Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States

Received  August 2009 Revised  November 2009 Published  January 2010

We prove several generic existence results for infinitely many periodic orbits of Hamiltonian diffeomorphisms or Reeb flows. For example, we show that a Hamiltonian diffeomorphism of a complex projective space or Grassmannian generically has infinitely many periodic orbits. We also consider symplectomorphisms of the two-torus with irrational flux. We show that a symplectomorphism necessarily has infinitely many periodic orbits if it has one and all periodic points are nondegenerate.
Citation: Viktor L. Ginzburg, Başak Z. Gürel. On the generic existence of periodic orbits in Hamiltonian dynamics. Journal of Modern Dynamics, 2009, 3 (4) : 595-610. doi: 10.3934/jmd.2009.3.595
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