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

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(M) (for example, if codim(M) > dim(M)), 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 [18, 19, 26]. We partially recover some existence results of Arnold [1] for Hamiltonian flows which describe a charged particle moving in a nondegenerate magnetic field on a torus. Following [43], we also relate our refined capacity to the study of Hamiltonian paths with minimal Hofer length.