On the minimizing measures of Lagrangian dynamical systems

The author considers dynamical systems generated by time-dependent periodic Lagrangians on a closed manifold M. An invariant probability mu of such a system has an homology rho ( mu ) in H₁(M,R) describing (roughly speaking) the average homological position of mu a.e. orbit, and an action A( mu ) defined as the integral with respect to mu of the Lagrangian. The minimizing measures are defined as the invariant probabilities of the system that minimize the action among those that have a given homology gamma . Their action beta ( gamma ) define a convex function beta :H₁(M,R) to R. These concepts were introduced by Mather where he proved several theorems about them. He further develops Mather's theory, giving a characterization of minimizing measures and proving properties about minimizing measures whose homologies are strictly extremal points of the function beta .