| In this thesis, we consider three applications of pseudoholomorphic curves to problems in Hamiltonian dynamics. In a first part, we prove an existence result for homoclinic orbits on a contact-type, critical energy level of an autonomous Hamiltonian, provided that the level is Hamiltonian displaceable. To do this, we transform the problem into a problem of Lagrangian intersection Floer theory. This involves a construction due to Mohnke [34] and some ideas from Legendrian surgery. In particular, we prove a generalization of Séré's result [36] on the existence of homoclinic orbits for an autonomous Hamiltonian system.;In a second part, we develop a theory of pseudoholomorphic curves into a singular contact manifold, which represents the critical level of an autonomous Hamiltonian. We show that a pseudoholomorphic half-plane is asymptotic to a homoclinic orbit. Furthermore, in a non-degenerate case, this convergence is of an exponential nature. This result is a first step towards understanding the change in contact homology under Legendrian surgery. Such a surgery formula would enable the computation of contact homology for every contact three manifold.;Finally, we lay the groundwork for an energy quantization result for pseudoholomorphic planes with a weaker energy bound than in the existing theory. This is a question related to the problem of understanding the compactification of the space of generalized pseudoholomorphic curves as envisioned by Hofer and studied by Abbas, Cieliebak and Hofer in [3]. This is part of an ongoing project with Casim Abbas and Helmut Hofer. |
