Squeezing in Floer theory and refined Hofer–Zehnder capacities of sets near symplectic submanifolds

Abstract

We use Floer homology to study the Hofer–Zehnder capacity of neighborhoods near a closed symplectic submanifold $M$ of a geometrically bounded and symplectically aspherical ambient manifold. We prove that, when the unit normal bundle of $M$ is homologically trivial in degree $dim\left(M\right)$ (for example, if $codim\left(M\right)>dim\left(M\right)$), a refined version of the Hofer–Zehnder capacity is finite for all open sets close enough to $M$. We compute this capacity for certain tubular neighborhoods of $M$ by using a squeezing argument in which the algebraic framework of Floer theory is used to detect nontrivial periodic orbits. As an application, we partially recover some existence results of Arnold for Hamiltonian flows which describe a charged particle moving in a nondegenerate magnetic field on a torus. Following an earlier paper, we also relate our refined capacity to the study of Hamiltonian paths with minimal Hofer length.