Finsler metrics and action potentials

We study the behavior of Mañé's action potential $\Phi_k$ associated to a convex superlinear Lagrangian, for $k$ bigger than the critical value $c(L)$. We obtain growth estimates for the action potential as a function of $k$. We also prove that the action potential can be written as $\Phi_k(x,y)=D_F(x,y)+f(y)-f(x)$ where $f$ is a smooth function and $D_F$ is the distance function associated to a Finsler metric.