Generalized Curie-Weiss-Potts model and quadratic pressure in ergodic theory

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 $+\mathrm{\infty }$.