  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 + dim(M)/2 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.
