We consider systems for which there exists decay of correlations against $L^1$ observables. Examples include expanding and piecewise expanding systems. For such systems we consider rare events consisting on the entrance into very small neighbourhoods of some chosen points $z$ on the phase space. We will see that there is a dichotomy regarding the extremal behaviour of these systems, depending on whether the point $z$ is periodic or not. Namely, we will see that if the point $z$ is periodic then we have an Extremal Index (EI) equal to 1 (which means no clustering of rare events) and the point processes counting the occurrence of rare events converge to a standard Poisson process. On the other hand, if the point $z$ is periodic we obtain an EI less than 1 (which means the occurrence of clustering) and the rare events point processes converge to a compound Poisson process.