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

TL;DR

This paper demonstrates that for generic exact magnetic Lagrangians on a closed manifold, the Mañé set properties are typical, with unique ergodic measures supported on hyperbolic periodic orbits exhibiting transversal intersections.

Contribution

It establishes generic properties of Mañé's sets for exact magnetic Lagrangians, including ergodicity and hyperbolicity of supporting measures, extending previous results to this specific class.

Findings

Existence of a residual set where Mañé, Aubry, and Mather sets coincide.

Generic support of unique ergodic measures on hyperbolic periodic orbits.

Transversal intersection of stable and unstable manifolds for these orbits.

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.