Generic Properties of Mañé’s Set of Exact Magnetic Lagrangians

Abstract

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 $${\widetilde{\mathcal{M}}}\left(c\right)={\widetilde{\mathcal{A}}}\left(c\right)={\widetilde{\mathcal{N}}}\left(c\right),\forall c\in\mathcal{G},$$ 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].