Extremal behaviour and rare events point processes for chaotic dynamical systems

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. We associate the existence of an Extremal Index less than 1 to the occurrence of periodic phenomena. For generic points, the exceedances, in the limit, are singular and occur at Poisson times. But around periodic points, the respective point processes of exceedances converge to a compound Poisson process. The extremal index usually coincides with the reciprocal of the mean of the limiting cluster size distribution. Here, we build dynamically generated stochastic processes with an extremal index for which that equality does not hold.