Extremal behaviour of chaotic dynamics

Abstract

We present a review of recent results regarding the existence of extreme value laws for stochastic processes arising from dynamical systems. We gather all the conditions on the dependence structure of stationary stochastic processes in order to obtain both the distributional limit for partial maxima and the convergence of point processes of rare events. We also discuss the existence of clustering which can be detected by an extremal index less than 1 and relate it with the occurrence of rare events around periodic points. We also present the connection between the existence of extreme value laws for certain dynamically defined stationary stochastic processes and the existence of hitting times statistics (or return times statistics). Finally, we make a complete description of the extremal behaviour of expanding and piecewise expanding systems by giving a dichotomy regarding the types of extreme value laws that apply. Namely, we show that around periodic points we have an extremal index less than 1 and at very other point we have an extremal index equal to 1.