Abstract. We consider Reeb flows on the tight 3-sphere admitting a pair of closed orbits forming a Hopf link. If the rotation numbers associated to the transverse linearized dynamics at these orbits fail to satisfy a certain resonance condition then there exist infinitely many periodic trajectories distinguished by their linking numbers with the components of the link. This result admits a natural comparison to the Poincaré-Birkhoff theorem on area-preserving annulus homeomorphisms. An analogous theorem holds on SO(3) and applies to geodesic flows of Finsler metrics on S 2 .

For Reeb vector fields on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other knotting/linking properties relative to the original set. We work out a few examples on the 3-sphere to illustrate the theory, and describe an application to closed geodesics on S 2 (a version of a result of Angenent in [Ang05]).

Abstract. We prove that on a restricted contact type hypersurface the number of leaf-wise intersections is bounded from below by a certain cup-length.