Journal of Modern Dynamics (JMD)

On the generic existence of periodic orbits in Hamiltonian dynamics

Pages: 595 - 610, Issue 4, October 2009      doi:10.3934/jmd.2009.3.595

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Viktor L. Ginzburg - Department of Mathematics, University of California at Santa Cruz, Santa Cruz, CA 95064, United States (email)
Başak Z. Gürel - Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States (email)

Abstract: 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.

Keywords:  Periodic orbits,Hamiltonian flows, Floer homology, Conley conjecture.
Mathematics Subject Classification:  Primary: 53D40; Secondary: 37J10.

Received: August 2009;      Revised: November 2009;      Available Online: January 2010.