On the graph theorem for Lagrangian invariant tori with totally irrational invariant sets

We show that every C 2 Lagrangian invariant torus W of a Tonelli Hamiltonian containing a uniformly continuous curve whose canonical projection has totally irrational homology is a graph, namely, the canonical projection restricted to W is a diffeomorphism. This result extends the graph property obtained by Bangert and Bialy–Polterovich for Lagrangian minimizing tori, invariant by the geodesic flow of a Riemannian metric in the 2-torus, without periodic orbits. Motivated by the famous Hedlund’s examples of Riemannian metrics in the n -torus with n closed, homology independent, minimizing geodesics having minimizing tunnels, we also show that Lagrangian, invariant tori with“large” homology (in the sense of Proposition 4.1) must be graphs. Moreover, we show the C 1 -generic nonexistence of Lagrangian invariant tori with“large”homology.