In 1986, V.I. Arnold formulated a conjecture which up to this day is remarkably little understood. It concerns the dynamics of a Reeb flow in relation to a Legendrian submanifold. More precisely, it says that for a horizontally displaceable Legendrian submanifold L of the contactization of a Liouville manifold, the number of Reeb chords on L is bounded from below by half of the Betti numbers of L. We will discuss several recent contributions to this conjecture and the way to prove its generalization when L admits an exact Lagrangian filling.
This is joint work with G. Dimitroglou Rizell.