Generating functions in symplectic and contact geometry

TL;DR

This paper explores generating functions in symplectic and contact geometry, focusing on translated points of contactomorphisms, and extends Sandon's contact Arnol'd conjecture proof by addressing key assumptions and gaps.

Contribution

It proves the contact Arnol'd conjecture under an additional generating function assumption and fills gaps in Sandon's original argument.

Findings

Proves the contact Arnol'd conjecture with a new assumption.

Completes the proof by addressing gaps in Sandon's approach.

Provides a refined understanding of translated points in 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 master's thesis proves the result under this additional assumption and fills minor gaps in other parts of Sandon's argument.