On a theorem due to Birkhoff

TL;DR

This paper proves that certain invariant submanifolds in the cotangent bundle of a compact, connected manifold are necessarily graphs, extending Birkhoff's theorem in Hamiltonian dynamics.

Contribution

It establishes that invariant submanifolds Hamiltonianly isotopic to the zero-section are graphs under Tonelli flow conditions, generalizing classical results.

Findings

Invariant submanifolds are graphs

Results apply to Tonelli flows

Extends Birkhoff's theorem

Abstract

The manifold M being compact and connected, we prove that every submanifold of its cotangent bundle that is Hamiltonianly isotopic to the zero-section and that is invariant by a Tonelli flow is a graph.