Many recent existence results for periodic orbits in Hamiltonian systems rely on proving the existence of a pseudoholomorphic curve. This existence problem, however, remains difficult. A problem still open is the Weinstein conjecture, which would assert that for any contact form on any contact manifold, the Reeb vector field has a closed orbit.
We will discuss a programme to solve the Weinstein conjecture in dimension three, introduced by Abbas, Cieliebak and Hofer. This approach combines the Giroux open book decomposition of a contact manifold with a new generalization of the notion of a pseudoholomorphic curve. I will present joint work with Abbas and Hofer on understanding the compactness properties of these curves, by means of a construction of "renormalization".