  A classical Borel Cantelli Lemma gives conditions for deciding whether an
infinite number of rare events will almost surely happen. In this article, we
propose an extension of Borel Cantelli Lemma to characterize the multiple
occurrence of events on the same time scale. Our results imply multiple
Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws
for systems which are exponentially mixing of all orders. The applications
include geodesic flows on compact negatively curved manifolds, geodesic
excursions, Diophantine approximations and extreme value theory for dynamical
systems.
