A note on the existence of U-cyclic elements in periodic Floer homology

Edtmair-Hutchings have recently defined, using Periodic Floer homology, a “U-cycle property” for Hamiltonian isotopy classes of area-preserving diffeomorphisms of closed surfaces. They show that every Hamiltonian isotopy class satisfying the U-cycle property satisfies the smooth closing lemma and also satisfies a kind of Weyl law involving the actions of certain periodic points; they show that every rational isotopy class on the two-torus satisfies the U-cycle property. The purpose of this note is to explain why the U-cycle property holds for every rational Hamiltonian isotopy class.