Generic Properties of Mañé's Set of Exact Magnetic Lagrangians

Let $M$ be a closed manifold and $L$ an exact magnetic Lagrangian. In this paper we prove that there exists a residual set $\mathcal{G}$ of $% H^{1}\left( M;\mathbb{R}\right)$ such that the property \begin{equation*} {\widetilde{\mathcal{M}}}\left( c\right) ={\widetilde{\mathcal{A}}}\left( c\right) ={\widetilde{\mathcal{N}}}\left( c\right), \forall c\in \mathcal{G}, \end{equation*} with ${\widetilde{\mathcal{M}}}\left( c\right)$ supporting a uniquely ergodic measure, is generic in the family of exact magnetic Lagrangians. We also prove that, for a fixed cohomology class $c$, there exists a residual set of exact magnetic Lagrangians such that, when this unique measure is supported on a periodic orbit, this orbit is hyperbolic and its stable and unstable manifolds intersect transversally. This result is a version of an analogous theorem, for Tonelli Lagrangians, proven in [6].