On the set of asymptotic homologies of orbits on invariant Lagrangian graphs

The paper proves that for a C2 invariant Lagrangian graph W on T^n lying on a supercritical level E>c0(L), all asymptotic homologies generated by quasi-orbits of the projected flow lie in a proper cone (Theorem 1.1) . It reduces the Tonelli case to a Riemannian geodesic flow on T^n (Proposition 3.2) and then constructs an explicit cone using stable-norm estimates for quasi-orbits (Definition 3.4 and Theorem 3.3) . The candidate solution reaches the same conclusion via Mather’s α–β theory: it identifies the graph with a fixed cohomology class c, shows α(c)=E and that all rotation vectors lie in ∂α(c), then derives a uniform positivity ⟨c,h⟩≥E−c0(L)>0 and encloses the asymptotic homologies in a proper cone. This is consistent with the paper’s setting (Appendix A on rotation vectors and minimizing measures) and with the supercritical hypothesis (Section 2.3) . The two arguments are different in technique (geometric foliation/metric estimates vs. convex duality), but there is no logical conflict; both establish the same theorem in the stated regime, and both note failure at the strict critical value via a counterexample (Section 3.4) .