Generic properties and problems of minimizing measures of Lagrangian systems

It is proved here that minimizing measures of a Lagrangian flow are invariant and the Lagrangian is cohomologous to a constant on the support of their ergodic components. Moreover, it is shown that generic Lagrangians have a unique minimizing measure which is uniquely ergodic and is a limit of invariant probabilities supported on periodic orbits of the Lagrangian flows.