  Stable Hamiltonian structures generalize contact forms and define a
volume-preserving vector field known as the Reeb vector field. We study two
aspects of Reeb vector fields defined by stable Hamiltonian structures on
3-manifolds: on one hand, we classify all the examples with finitely many
periodic orbits under a non-degeneracy condition; on the other, we give
sufficient conditions for the existence of a supporting broken book
decomposition and for the existence of a Birkhoff section.
