Abstract

A section to a flow in a closed 3-manifold, is a transverse surface that allows to understand the dynamics of the flow through a diffeomorphism of the surface. Not all the flows admit sections, the question motivating the talk is when a Reeb vector field admits either a Birkhoff section or a finite collection of transverse surfaces coding the dynamics. Reeb vector fields, associated to contact structures, are an important family of volume preserving vector fields on closed 3-manifolds.

For Reeb vector fields, Giroux's correspondance gives, in particular, for every contact structure on a closed 3-manifold an adapted open book decomposition. Hence, there exists a Reeb vector that is tangent to the binding and transverse to the interior of the pages. For this vector field, each page is a Birkhoff section and the dynamics of the flow can be studied from the first return map. This correspondance is unsatisfactory when one wants to study all the Reeb vector fields associated to a contact structure.

In collaboration with V. Colin and P. Dehornoy, we proved that every non-degenerate Reeb vector field on a closed 3-manifold is adapted to a broken book (a generalisation of open book). This construction gives a system of transverse surfaces with boundary and allows to establish results on the dynamics of the vector field. For example, we proved that any non-degenerate Reeb vector field has either two or infinitely many periodic orbits. I will explain other application of broken books decompositions.