2024/09/24 by Ruggiero, Rafael Oswaldo, Sorrentino, Alfonso
#Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2409.16010
Given a smooth Tonelli Hamiltonian on the torus \mathbbTn and a C2 Lagrangian graph W ⊂ T*\mathbbTn 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 H1(\mathbbTn,ℝ) that contains the asymptotic homologies of the canonical projections of recurrent orbits in W. Additionally, for invariant Lagrangian graphs on \mathbbT3, drawing on Franks' theory of the rotation set of homeomorphisms of \mathbbT2 homotopic to the identity, we show that under certain assumptions for an invariant Lagrangian graph on \mathbbT3, if there exists a rational vector in homology contained in the set of asymptotic homologies of orbits on 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 \mathbbT2. 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.