  Hidden Markov Processes (HMP) is one of the basic tools of the modern
probabilistic modeling. The characterization of their entropy remains however
an open problem. Here the entropy of HMP is calculated via the cycle expansion
of the zeta-function, a method adopted from the theory of dynamical systems.
For a class of HMP this method produces exact results both for the entropy and
the moment-generating function. The latter allows to estimate, via the Chernoff
bound, the probabilities of large deviations for the HMP. More generally, the
method offers a representation of the moment-generating function and of the
entropy via convergent series.
