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

Abstract Given a smooth Tonelli Hamiltonian on the torus $$\mathbb {T}^{n}$$ T n and a $$C^{2}$$ C 2 Lagrangian graph $$W \subset T^{*}\mathbb {T}^{n}$$ W ⊂ T ∗ T n that is invariant under the Hamiltonian flow and contained within a Mañé supercritical energy level, we demonstrate the existence of a proper cone in the first real homology group $$H_1({\mathbb {T}}^n,\mathbb {R})$$ H 1 ( T n , R ) that contains the asymptotic homologies of the canonical projections of recurrent orbits in W . Additionally, for invariant Lagrangian graphs on $$\mathbb {T}^{3}$$ T 3 , drawing on Franks’ theory of the rotation set of homeomorphisms of $$\mathbb {T}^{2}$$ T 2 homotopic to the identity, we show that if there exists a rational vector in homology contained in the interior of the set of asymptotic homologies of orbits of the Lagrangian graph, then the graph contains a Mather measure supported on a periodic orbit. This result generalizes a well-known fact for Lagrangian graphs on $$\mathbb {T}^{2}$$ T 2 . Finally, we exploit these results for three dimensional tori to give a partial answer to a conjecture by Carneiro–Ruggiero about the non-existence of Hedlund Lagrangian tori at supercritical energy levels.