| Quasi-periodic Lagrangian systems on the annulus are studied in this thesis. We show that such a system is a natural rendezvous of Aubry Mather theory and its higher dimensional counterpart. The goal is to study the set of globally action minimizing orbits and construct new orbits for the purpose of connecting. In particular, we prove that the Aubry crossing lemma holds for the infinite sequence of generating functions of the system. As a consequence of this important technical lemma, strict convexity of the β function in the sense of Mather can be proved. Therefore, given any rotation number, there exist action minimizing orbits of that rotation number. It then becomes clear that the pathology of Hedlund's example does not occur for quasi-periodic systems on the annulus.; The second part of the thesis is devoted to proving that diffusion takes place when there is no invariant two torus in the phase space. Codimension one invariant tori are the weakest obstruction to the existence of diffusive orbit, it turns out it is sufficient to prove diffusion when such obstruction is absent. In this regard, locally action minimizing orbits are carefully constructed. The difficulty in this construction is to choose appropriate constraints for the minimization and prove that these orbits do not bump up against the constraints. Globally action minimizing sets are weakly hyperbolic in the sense that they have stable and unstable sets which are locally action minimizing. The culmination of our constructions is the proof of a shadowing theorem without explicit hyperbolicity assumptions on the system. |
