Limits of equilibrium states for coupled weakly interacting systems. Application to the measure of maximal entropy

We study metastability for symbolic dynamic. We prove that for a global system given by two independent sub-systems linked by a hole, and for a Lipschitz continuous potential, the global equilibrium state converges, as the hole shrinks, to a convex combination of the two independent equilibria in each component. Two kinds of convergence occur, depending on the assumptions on how long an orbit has to stay in each well. As a by-product, we show that this can be applied to a geometrical system inspired from [11] and for the measure of maximal entropy.