  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.
