Cluster distributions for dynamically defined point processes

Abstract

The emergence of clustering of rare events for chaotic dynamics was first observed as a consequence of periodicity, i.e., by considering target sets that shrink to a periodic point, one was able to create fast returns to these target sets which were responsible for the appearance of a bulk of high observations of observable functions that were maximised at the periodic point. This meant that the Rare Events Point Processes, counting the number of entrances in these target sets, converge to a compound Poisson process, with a geometric multiplicity distribution ruling the cluster sizes. In Azevedo et al. (2016), a new mechanism to create clustering of rare events was introduced by considering observable functions maximised at a finite number of points that were linked by belonging to the same orbit. We make a deep study of the potential of this mechanism to produce different multiplicity distributions. Namely, we show that with the right choice of a system and observable, one can obtain any given finitely supported cluster size distribution. We also study the impact of symmetry and other properties of the systems on the possible clustering size distributions, which are also classified for the case of periodic maximal orbits.