Over a -mixing dynamical system we consider the function (Cn)/n in the limit of large n, where (Cn) is the first return of a cylinder of length n to itself. Saussol et al. ([30]) proved that this function is constant almost everywhere if the Cn 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 R'enyi's entropy of the system. We finally compute the free energy function of (Cn)/n. We illustrate our results with a few examples.