Appears in collection : Giroux 60 - Convexity in contact and symplectic topology

A Birkhoff section reduces the study of a non-singular flow in 3D to that of a surface diffeomorphism and provides a rational open book carrying the flow. I will present a recent existence statement for Birkhoff sections: the set of a Reeb vector fields on closed 3-manifolds that admit a Birkhoff section contains an open and dense subset in the 𝐶∞ topology. This construction is based on the existence of broken book decompositions and is part of a joint work with Vincent Colin, Pierre Dehornoy and Umberto Hryniewicz.