Over a psi-mixing dynamical system we consider the function tau(C-n)/n in the limit of large n, where tau(C-n) is the first return of a cylinder of length n to itself. Saussol et al. ([ ]) proved that this function is constant almost everywhere if the C-n are chosen in a descending sequence of cylinders around a given point. We prove upper and lower general bounds for its large deviation function. Under mild assumptions we compute the large deviation function directly and show that the limit corresponds to the Renyi's entropy of the system. We finally compute the free energy function of tau(C-n)/n. We illustrate our results with a few examples.213729747