The dynamics of magnetic flows for energies above Mane's critical value.

We show that, for energies above Mane's critical value, minimal magnetic geodesics are Riemannian (A,0)-quasi-geodesics where A -> 1 as the energy tends to infinity. As a consequence, on negatively curved manifolds, minimal magnetic geodesics lie in tubes around Riemannian geodesics. Finally, we investigate a natural metric introduced by Mane via the so-called action potential. Although this magnetic metric does depend on the magnetic field, the associated magnetic length turns out to be just the Riemannian length