  We extend results on quadratic pressure and convergence of Gibbs mesures from
previous joined work of the authors to the Curie-Weiss-Potts model. We define
the notion of equilibrium state for the quadratic pressure and show that under
some conditions on the maxima for some auxiliary function, the Gibbs measure
converges to a convex combination of eigen-measures for the Transfer Operator.
This extension works for dynamical systems defined by {infinite-to-one} maps.
As an example, we compute the equilibrium for {the mean-field} $XY$ model as
the number of particles goes to $+\infty$.
