We consider a continuous coercive Hamiltonian on the cotangent bundle of the compact connected manifold M which is convex in the momentum. We prove that the viscosity solutions uλ:M→R of the critical Hamilton-Jacobi equation with discount factor λ>0 converge uniformly, as λ goes to 0, to a specific solution u0:M→R of the limit equation. We characterize u0 in terms of Peierls barrier and projected Mather measures. As a corollary, we infer that the ergodic approximation, as introduced by Lions, Papanicolaou and Varadhan in 1987 in their seminal paper on periodic homogenization of Hamilton-Jacobi equations, selects a specific corrector in the limit. The talk is based on a joint work with A. Fathi, R. Iturriaga and M. Zavidovique that will appear on Inventiones Mathematicae.