We discus the existence of a unique, exact and acip probability measure when the system under consideration is Gibbs-Markov. We prove the existence of the measure when the system itself is not Gibbs-Markov but there exists an induced Gibbs-Markov system. Also we prove the decay of correlation with respect to that measure using Young Towers.
Finally we prove that, for uniformly expanding circle map, any interval of the circle exhibit a Gibbs-Markov structure which allows us to obtain estimates for the rate of mixing for Holder continuous observable.