We study $(h)$ -minimal configurations in Aubry-Mather theory, where $h$ belongs to a complete metric space of functions. Such minimal configurations have definite rotation number. We establish the existence of a set of functions, which is a countable intersection of open everywhere dense subsets of the space and such that for each element $h$ of this set and each rational number $\alpha$ , the following properties hold: (i) there exist three different $(h)$ -minimal configurations with rotation number $\alpha$ ; (ii) any $(h)$ -minimal configuration with rotation number $\alpha$ is a translation of one of these configurations.