Rare events for product fractal sets * *All authors were partially supported by by FCT Project PTDC/MAT-PUR/28177/2017 and by CMUP, which is financed by national funds through FCT-Fundação para a Ciência e a Tecnologia, IP, under the project with reference UIDB/00144/2020.

Abstract

We analyse the existence of limiting laws of rare events for dynamically generated stochastic processes. We consider two-dimensional dynamical systems and observable functions maximised on Cantor dust sets and prove the existence of distributional limits for the partial maxima. We show how the extremal index, measuring the degree of clustering of rare events, is linked to the compatibility between the dynamics and the fractal structure of the maximal sets.