Homotopical Minimal Measures for Geodesic flows on Surfaces of Higher Genus

Abstract

We study the homotopical minimal measures for positive definite autonomous Lagrangian systems. Homotopical minimal measures are action-minimizers in their homotopy classes, while the classical minimal measures (Mather measures) are action-minimizers in homology classes. Homotopical minimal measures are much more general, they are not necessarily homological action-minimizers. 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 $\phi_t: TM\rightarrow TM$ is exactly the complete geodesic flow on $TM$. We show that for each homotopical minimal ergodic measure $\mu$ that is supported on a nontrivial simple closed periodic trajectory, there is a finite-fold covering space $M'$ such that each ergodic preimage of $\mu$ on $TM'$ is a minimal measure in the classic Mather theory for the Lagrangian system on $TM'$.