Fixed points of symplectic periodic flows

Alvaro Pelayo, Susan Tolman

Research output: Contribution to journalArticlepeer-review


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 manifold M, then it is classically known that there are at least 1/2 dim M + 1 fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah-Bott-Berline-Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective - the Chern class map assigns to a fixed point the sum of the action weights at the point. We complement this result with less sharp lower bounds on the number of fixed points, under no assumptions; from a dynamical systems viewpoint, our results imply that there is no symplectic periodic flow with exactly one or two equilibrium points on a compact manifold of dimension at least eight.

Original languageEnglish (US)
Pages (from-to)1237-1247
Number of pages11
JournalErgodic Theory and Dynamical Systems
Issue number4
StatePublished - Aug 2011

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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