  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.
