Hyperbolicity and exponential long-time convergence for space-time periodic Hamilton-Jacobi equations

We prove exponential convergence to time-periodic states of the solutions of time-periodic Hamilton-Jacobi equations on the torus, assuming that the Aubry set is the union of a finite number of hyperbolic periodic orbits of the the Euler Lagrange flow. The period of limiting solutions is the least common multiple of the periods of the orbits in the Aubry set. This extends a result that we obtained in the autonomous case.