Abstract: The aim of this talk is to describe a work in progress about the limit of equilibrium states for certain potentials on the shift. The potential chosen reaches its maximum at exactly two distinct periodic points, but its local behaviors are distinct; based on this information we can find the limit measure when the potential $V$ is replaced by $\beta V$ and we take $\beta$ going to infinity. This is a joint work with R. Leplaideur and A. Lopes.