  We study the set of invariant idempotent probabilities for place dependent
idempotent iterated function systems defined in compact metric spaces. Using
well-known ideas from dynamical systems, such as the Ma\~{n}\'{e} potential and
the Aubry set, we provide a complete characterization of the densities of such
idempotent probabilities. As an application, we provide an alternative formula
for the attractor of a class of fuzzy iterated function systems.
