Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry-Mather theory

TL;DR

This paper develops graph selectors for exact Lagrangians in cotangent bundles, linking spectral invariants and Fukaya categories to characterize Aubry-Mather sets and invariant Lagrangians in Hamiltonian dynamics.

Contribution

It introduces Lipschitz-exact Lagrangians and extends graph selector techniques, providing new insights into Aubry-Mather theory and Hamiltonian invariants.

Findings

Constructed graph selectors for compact exact Lagrangians.

Characterized Aubry and Mather sets using spectral invariants.

Generalized invariance results for Lagrangians under Tonelli Hamiltonian flows.

Abstract

We construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants developed by Oh and results by Abouzaid about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalization of graph selector. We then, following Bernard-Oliveira dos Santos, use these results to give a new characterization of the Aubry and Mane sets of a Tonelli Hamiltonian and to generalize a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.