  Weak KAM theory for discount Hamilton-Jacobi equations and corresponding
discount Lagrangian/Hamiltonian dynamics is developed. Then it is applied to
error estimates for viscosity solutions in the vanishing discount process. The
main feature is to introduce and investigate the family of $\alpha$-limit
points of minimizing curves, with some details in terms of minimizing measures.
In error estimates, the family of $\alpha$-limit points is effectively
exploited with properties of the corresponding dynamical systems.
