# Rare events for Cantor target sets

## TL;DR

This paper investigates the limiting laws of rare events related to orbits entering target sets shrinking to a Cantor set, analyzing the role of clustering and the Extremal Index in fractal dynamics.

## Contribution

It introduces a method to compute the Extremal Index for Cantor sets, linking it to the fractal geometry and dynamics of the system.

## Key findings

- Extremal Index helps identify compatibility between dynamics and fractal structure.
- The Extremal Index is connected to the box dimension of intersections.
- Limiting laws are established for rare events involving Cantor target sets.

## Abstract

We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a Cantor set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal structure of the limiting Cantor set. The computation of the Extremal Index is connected to the box dimension of the intersection between the Cantor set and its iterates.