Foliations forcing closed orbits

Knowing topological and geometric properties about foliations on hyperbolic 3-manifolds often yields dynamical information about transverse flows, and vice versa. In this vein, we address the following question: What properties of a foliation force all transverse flows to have closed orbits? We answer this question for finite depth foliations, a class of foliations generalizing finite-type fibrations by allowing a hierarchy of spiraling behavior. We will focus on the depth one case for concreteness, showing roughly that foliations of “intermediate complexity” admit transverse flows without closed orbits, while the most simple and most complicated foliations force closed orbits. This is joint work in progress with Audrey Rosevear.