Extremal behaviour of deterministic and random dynamical systems

It is well known that the Extremal Index (EI) measures the intensity of clustering of extreme events in stationary processes. We sill see that for some certain uniformly expanding systems there exists a dichotomy based on whether the rare events correspond to the entrance in small balls around a periodic point or a non-periodic point. In fact, either there exists EI in (0,1) around (repelling) periodic points or the EI is equal to 1 at every non-periodic point. The main assumption is that the systems have sufficient decay of correlations of observables in some Banach space against all L¹-observables.

Then we consider random perturbations of uniformly expanding systems, such as piecewise expanding maps of the circle. We will see that, in this context, for additive absolutely continuous noise (w.r.t. Lebesgue), the dichotomy vanishes and the EI is always 1.