Tonelli Lagrangian systems on the 2-torus and topological entropy

We study Tonelli Lagrangian systems on the 2-torus in energy levels above Ma\~n\'e's strict critical value and analyize the structure of global minimizers in the spirit of Morse, Hedlund and Bangert. In the case where the topological entropy of the Euler-Lagrange flow on the fixed energy level vanishes, we show that there are invariant tori for all rotation vectors indicating integrable-like behavior on a large scale. On the other hand, using a construction of Katok, we give examples of reversible Finsler geodesic flows with vanishing topological entropy, but having ergodic components of positive measure in the unit tangent bundle.