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

TL;DR

This paper proves a multidimensional Birkhoff theorem for recurrent Lagrangian submanifolds under Tonelli Hamiltonian flows, establishing conditions under which such submanifolds are graphs over the zero section.

Contribution

It extends Birkhoff's theorem to a multidimensional setting for recurrent Lagrangian submanifolds in Hamiltonian dynamics with new convergence criteria.

Findings

Recurrent Lagrangian submanifolds are graphs over the zero section.

Convergent subsequences in both time directions imply graph structure.

Lagrangian submanifolds exhibit recurrence under specific conditions.

Abstract

Consider a closed manifold $M$ and a time-periodic Tonelli Hamiltonian $H : \mathbb{R}/\mathbb{Z} \times T^*M \to \mathbb{R}$ with flow $\phi_H$. Let $\mathcal{L} \subset T^*M$ 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 $0_{T^*M}$ of $T^*M$. Furthermore, we show that $\mathcal{L}$ is recurrent in both positive and negative times for the same type of convergence.