Tracking trajectories in Hamiltonian systems using holomorphic curve tools.

The goal is to describe how techniques from symplectic dynamics can be used to study orbit travel in three dimensions, for systems like the restricted 3-body problem from celestial mechanics. The pseudo-holomorphic curve theory initiated by Hofer, Wysocki and Zehnder gives a decomposition of the space into regions whose boundaries are surfaces transverse to the flow. (Pseudo-holomorphic curves are special minimal surfaces). One can label the regions A, B, C etc and form a directed graph. This gives us a natural language to discuss trajectories of orbits, a topic notorious for its complexity. Certain interesting features arise from this structure. I will describe how semi-local considerations lead to more global information and symbolic dynamics. This is joint work with Umberto Hryniewicz and Gerhard Knieper. The talk will address a general mathematical audience.