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The Ruelle thermodynamic formalism for dynamical trajectories over the largetime $T$ corresponds to the large deviation theory for the information per unittime of the trajectories probabilities. The microcanonical analysis consists inevaluating the exponential growth in $T$ of the number of trajectories with agiven information per unit time, while the canonical analysis amounts toanalyze the appropriate non-conserved $\beta$-deformed dynamics in order toobtain the scaled cumulant generating function of the information, the firstcumulant being the famous Kolmogorov-Sinai entropy. This framework is describedin detail for discrete-time Markov chains and for continuous-time Markov jumpprocesses converging towards some steady-state, where one can also constructthe Doob generator of the associated $\beta$-conditioned process. Theapplication to the Directed Random Trap model on a ring of $L$ sites allows toillustrate this general framework via explicit results for all the introducednotions. In particular, the glassy phase is characterized by anomalous scalinglaws with the size $L$ and by non-self-averaging properties of theKolmogorov-Sinai entropy and of the higher cumulants of the trajectoryinformation.

Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models