  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.
