Fixed points of symplectic periodic flows

By 'Alvaro Pelayo, S. Tolman. Ergodic Theory and Dynamical Systems, 2010

51 citations recorded by the source corpus.

Abstract 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 $\frac {1}{2}\,{\dim M}+1$ 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a545d96532c754a9ec2e0e029936ec1823924c51

Citations in the corpus

• Symplectic theory of completely integrable Hamiltonian systems (2011, unknown): “2], Pelayo and Tolman [86] for additional results in the case of compact manifolds, and Pelayo and Ratiu [84] for results in the case of noncompact manifolds.”