We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The Extremal Index, which is responsible for the appearance of clusters of exceedances, usually coincides with the reciprocal of the mean of the limiting cluster size distribution. In this talk, we show how to build dynamically generated stochastic processes with an Extremal Index for which that relation does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at at least two points of the phase space, where one of them is an indifferent periodic point and another one is either a repelling periodic point or a non periodic point. This enables to mix the behaviour of an Extremal Index equal to 0 with that of an Extremal Index larger than 0.
Ana Cristina Freitas