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Fixed points of symplectic periodic flows

Published online by Cambridge University Press:  16 June 2010

ALVARO PELAYO and
SUSAN TOLMAN
ALVARO PELAYO
Affiliation:
University of California–Berkeley, Mathematics Department, 970 Evans Hall #3840, Berkeley, CA 94720-3840, USA (email: apelayo@math.berkeley.edu)
SUSAN TOLMAN
Affiliation:
UI Urbana-Champaign, Department of Mathematics, 1409 W Green St, Urbana, IL 61801, USA (email: stolman@math.uiuc.edu)
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Abstract

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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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011, pp. 1237 - 1247
Copyright
Copyright © Cambridge University Press 2010

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