A Multidimensional Birkhoff Theorem for Recurrent Lagrangian Submanifolds by a Tonelli Hamiltonian

Consider a closed manifold $M$ and a time-periodic Tonelli Hamiltonian $H:\mathbb{R}\mathrm{/}\mathbb{Z}\times T^{\ast }M\to \mathbb{R}$ with flow ${\phi }_H$. Let $\mathcal{L}\subset T^{\ast }M$ be a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if ${\phi }_H^n(\mathcal{L})$ admits convergent subsequences in both positive and negative times, in the Hausdorff topology and with control on the Liouville primitives, to two Lagrangian submanifolds, then $\mathcal{L}$ is a graph over the zero section $0_{T^{\ast }M}$ of $T^{\ast }M$. Furthermore, we show that $\mathcal{L}$ is recurrent in both positive and negative times for the same type of convergence.