Remarks on symplectic circle actions, torsion and loops

Abstract

We study loops of symplectic diffeomorphisms of closed symplectic manifolds. Our main result, which is valid for a large class of symplectic manifolds, shows that the flux of a symplectic loop vanishes whenever its orbits are contractible. As a consequence, we obtain a new vanishing result for the flux group and new instances where the presence of a fixed point of a symplectic circle action is a sufficient condition for it to be Hamiltonian. We also obtain applications to symplectic torsion, more precisely, non-trivial elements of $\mathrm{Symp}_{0}(M,\omega)$ that have finite order.