  The thermodynamic formalism, which was first developed for dynamical systems
and then applied to discrete Markov processes, turns out to be well suited for
continuous time Markov processes as well, provided the definitions are
interpreted in an appropriate way. Besides, it can be reformulated in terms of
the generating function of an observable, and then extended to other
observables. In particular, the simple observable $K$ giving the number of
events occurring over a given time interval turns out to contain already the
signature of dynamical phase transitions. For mean-field models in equilibrium,
and in the limit of large systems, the formalism is rather simple to apply and
shows how thermodynamic phase transitions may modify the dynamical properties
of the systems. This is exemplified with the q-state mean-field Potts model,
for which the Ising limit q=2 is found to be qualitatively different from the
other cases.
