Vincent Colin, Nantes 

Title: On the dynamics of Reeb vector fields

Abstract: The study of “Reeb” vector fields on contact manifolds encompassesseveral classical situations such as the geodesic flow on a Riemannian manifold or the three-body problem in celestial mechanics. As early as the 1890s, Poincaré relied on the existence of surfaces of sections, now called “Birkhoff sections”, to study the dynamical properties of vector field flows, for instance the existence of periodic orbits. The use of pseudo-holomorphic curves techniques in contact geometry has recently made it possible to show the generic existence of such sections for Reeb fields on 3-dimensional contact manifolds and to deduce new results: on a closed 3-manifold, every non-degenerate Reeb field has either two or infinitely many periodic orbits, and two orbits are only possible on the sphere or a lens space. The degenerate case is now also addressed in many situations. In this talk, we will present an overview of the questions and tools involved in this new variation on the slogan “three implies chaos.”