Abstract

In this talk, we study contact structures that admit certain types of singularities, called $b^m$-contact structures. Those structures can be viewed as a particular case of Jacobi manifolds satisfying some transversality conditions. The motivation to study this generalization of contact structures arises from classical examples in celestial mechanics, as for example the restricted planar circular three body problem (RPC3BP), but also appears in the study of fluid dynamics on manifolds with cylindrical ends. We will focus on understanding the dynamics of the associated Reeb vector field of $b^m$-contact forms. Due to singularities, the dynamics are fundamentally different to smooth Reeb dynamics and we will discuss a singular version of Weinstein conjecture on the existence of periodic orbits on those manifolds. Time permitting, we talk about generic existence of so called singular periodic orbits. This is joint work with Eva Miranda and work in progress with Eva Miranda and Daniel Peralta-Salas.