Citation/Abstract

In this dissertation we study some generic properties of Lagrangian Systems and symplectic diffeomorphisms.

In Chapter 3 we give a negative answer to a conjecture proposed by Mañé, we give an example of a C ¹ open set U of Lagrangians on the n -torus such that for any Lagrangian in U there exist a cohomology class c ∈ H ¹ ([Special characters omitted.] ) which has at least n different ergodic c -minimal measures. The minimizing measures are used in the construction of orbits connecting different regions in the phase space (Arnold diffusion), so understanding their structure should be helpful.

We also consider symplectic diffeomorphisms on compact manifolds. We prove in Chapter 4 that if a symplectic diffeomorphism is not partially hyperbolic then with an arbitrarily small C ¹ perturbation we can create a totally elliptic periodic point inside any given open set on the manifold. As a consequence, a C ¹ generic symplectic diffeomorphism is either partially hyperbolic or it has dense elliptic periodic points.

From this theorems we obtain some interesting corollaries: (1) Any C ¹ robustly transitive symplectic diffeomorphism must be partially hyperbolic. (2) Any stably ergodic symplectic diffeomorphism must be partially hyperbolic.

The second corollary is a converse of the Pugh-Shub conjecture on stable ergodicity for the symplectic case. Here a map f is stably ergodic if there exist a C ¹ open neighborhood U of f such that every C ² map in U is ergodic.