Action Selectors and the Fixed Point Set of a Hamiltonian Diffeomorphism

Abstract

In this paper we study the size of the fixed point set of a Hamiltonian diffeomorphism on a closed symplectic manifold which is both rational and weakly monotone. We show that there exists a non-trivial cycle of fixed points whenever the action spectrum is smaller, in a certain sense, than required by the Ljusternik-Schirelman theory. For instance, in the aspherical case, we prove that when the number of points in the action spectrum is less than or equal to the cup length of the manifold, then the cohomology of the fixed point set must be non-trivial. This is a consequence of a more general result that is applicable to all weakly monotone manifolds asserting that the same is true when the action selectors are related by an equality of the Ljusternik-Schirelman theory.