Homotopical minimal measures for geodesic flows on surfaces of higher genus

We study the homotopical minimal measures for positive definite autonomous Lagrangian systems. Homotopical minimal measures are supported by trajectories with minimal action in their homotopy classes, while the classical minimal measures (Mather measures) are supported by action-minimizers in homology classes. Homotopical minimal measures are much more general, they are not necessarily Mather measures. However, some of them can be obtained from the classical ones by lifting them to finite-fold covering spaces. We apply this idea of finite covering to the geodesic flows on surfaces of higher genus. Let ( M , G ) be a compact closed surface with genus g > 1, where G is a complete Riemannian metric on M . Consider the positive definite autonomous Lagrangian L ( x , v ) = G x ( v , v ) , whose Lagrangian system ϕ t : T M → T M is exactly the complete geodesic flow on TM . We show that for each homotopical minimal ergodic measure µ that is supported on a nontrivial simple closed periodic trajectory, there is a finite-fold covering space M ′ such that each ergodic preimage of µ on TM ′ is a minimal measure in the classic Mather theory for the Lagrangian system on TM ′.