The paper deals with existence of a stationary Mather measure which is invariant under the Euler-Lagrange flow and is supported on a graph. Precisely, the authors consider Lagrangians \(L: {\mathbb R}^n\times {\mathbb R}^n\times\Omega\to{\mathbb R},\) where \(\Omega\) is a compact metric space on which \({\mathbb R}^n\) acts through an action which leaves \(L\) invariant.
The stationary Mather problem consists in minimizing the integral \[ \int_{{\mathbb R}^n\times\Omega} L(0,v,\omega)\;d\mu(v,\omega) \] over all probability measures satisfying the so-called holonomy constraint, and the minimizer is called a stationary Mather measure.
The authors generalize the standard Mather problem for quasi-periodic and almost-periodic Lagrangians and prove existence of stationary Mather measures invariant under the Euler-Lagrange flow. Several estimates are also obtained for viscosity solutions of the Hamilton-Jacobi equations for the discounted cost infinite horizon problem.