Abstract: This project investigates minimal geodesics on complete Riemannian manifolds with non-compact universal cover, in particular their ergodic properties w.r.t. naturally defined invariant measures. We shall also deal with twist maps featuring non-periodic angles, which exclude the application of KAM theory or Aubry-Mather theory. However, the dynamics of the minimizing orbits are deeply connected to those described in classic results of Hedlund for the 2-torus. Hence we plan to transfer the techniques from the non-periodic angles setting to establish structural phenomena of minimal geodesics (like recurrence) on non-compact cylinders, for instance.