  We define a nonlinear thermodynamical formalism which translates into
dynamical system theory the statistical mechanics of generalized mean-field
models, extending investigation of the quadratic case by Leplaideur and
Watbled.
  Under suitable conditions, we prove a variational principle for the nonlinear
pressure and we characterize the nonlinear equilibrium measures and relate them
to specific classical equilibrium measures.
  In this non-linear thermodynamical formalism, which can, e.g., model
mean-field approximation of large systems, several kind of phase transitions
appear, some of which cannot happen in the linear case. We use our
correspondence between non-linear and linear equilibrium measures to further
the understanding of phase transitions, { both in previously known cases
(Curie-Weiss and Potts models) and in} new examples (metastable phase
transition).
  Finally, we apply some of the ideas introduced to the classical
thermodynamical formalism, proving that freezing phase transitions can occur
over \emph{any} zero-entropy invariant compact subset of the phase space.
