Generating Functions in Symplectic and Contact Geometry

Abstract

A translated point of a contactomorphism $\phi$ on a contact manifold with contact form $\alpha$ is a point $p$ where $\alpha$ is preserved under $\phi$ and whose image under $\phi$ lies in the same Reeb trajectory. They were introduced as a contact analogon for fixed points of Hamiltonian diffeomorphisms by Sheila Sandon and can be understood as a special case of leafwise fixed points. She established a contact version of the non-degenerate Arnol'd conjecture on spheres using a generating function approach. It turns out that Sandon's proof only works under the assumption that there exists a generating function whose sublevel set at zero has nontrivial homology. This thesis proves the result under this additional assumption and fills gaps in other parts of Sandon's argument.