Reeb dynamics in dimension 3 and broken book decompositions

The study of the dynamics of vector fields in dimension 3 was at center of Poincaré’s pioneering work on the 3-body problem. Vector fields arising from mechanics often have the extra property of preserving a contact structure and fall in the family of Reeb vector fields. I will describe how to extend the notion of a Poincaré's surface of section to the more general notion of a broken book decomposition and derive general dynamical properties. Amongst those, a nondegenerate Reeb vector field has either 2 or infinitely many periodic orbits. This is joint work with Patrick Dehornoy and Ana Rechtman.