Generic properties of Ma\~n\'e's set of exact magnetic Lagrangians

Abstract

Let $M$ be a closed manifold and $L$ an exact magnetic Lagrangian. In this paper we proved that there exists a residual $\mathcal{G}$ 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 ${\widetilde{\mathcal{M}}}\left( c\right)$ supports on a uniquely ergodic measure, is generic in the family of exact magnetic Lagrangians. We also prove that, fixed the 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 Theorem D of \cite{gon5} for the exact magnetic Lagrangian case.