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

Abstract

Given a smooth Tonelli Hamiltonian on the torus ${\mathbb{T}}^n$ and a $C^2$ Lagrangian graph $W\subset T^{\ast }{\mathbb{T}}^n$ 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 $H_1({\mathbb{T}}^n,\mathbb{R})$ that contains the asymptotic homologies of the canonical projections of recurrent orbits in $W$. Additionally, for invariant Lagrangian graphs on ${\mathbb{T}}^3$, drawing on Franks' theory of the rotation set of homeomorphisms of ${\mathbb{T}}^2$ homotopic to the identity, we show that under certain assumptions for an invariant Lagrangian graph on ${\mathbb{T}}^3$, 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 ${\mathbb{T}}^2$. 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.

32 pages, 2 figures