After having
shown how the thermodynamic formalism can be applied to continuous time
Markov processes, we shall cast the latter formalism within the borader
framework of large deviations, thus establishing a bridge towards the
fluctaion theorem. We shall be interested in a much simpler than usual
observable, namely the number of events that have occurred in a fixed
time interval, and we will show that the distribution of the latter
observable contains the signature of dynamical phase transitions in
systems in or out of equilibrium. Finally, we will introduce the
tools allowing us to characterize dynamic phase transitions in terms of
phase separation within the space of trajectories. We will illustrate
our approach with concrete examples (random walks, exclusion processes,
Ising model).