On fluctuations and the exponential statistics of return times

This paper presents some facts related to the exponential statistics of ret urn times. We first investigate the question of computing the speed of conv ergence to this limiting law. We show that this speed carries some informat ion concerning the system under consideration, while via a local analysis w e can relate it to some combinatorial property of some orbits. Next, we pro ve that for an arbitrary dynamical system, the existence of an exponential statistic for the return times implies the equivalence between the fluctuat ions of the empirical entropies and the repetition times. AMS classificatio n scheme numbers: 37B20, 37D25, 37D50, 60F05, 94A17.