Invariant tori and topological entropy in Tonelli Lagrangian systems on the 2-torus

摘要

We study the Euler-Lagrange flow of a Tonelli Lagrangian on the 2-torus T-2 at a fixed energy level epsilon subset of TT2 strictly above Mane's strict critical value. We prove that, if for some rational direction zeta is an element of S-1 there is no invariant graph T subset of epsilon over T-2 for the Euler-Lagrange flow with the property that all orbits on T have an asymptotic direction equal to zeta, then there are chaotic dynamics in epsilon. This implies that, if the topological entropy of the Euler-Lagrange flow in epsilon vanishes, then in epsilon there are invariant graphs for all asymptotic directions zeta is an element of S-1 and integrable-like behavior on a large scale.