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
\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].
Keywords:exact magnetic Lagrangian, Mañé set, genericity.
References