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A classical Borel Cantelli Lemma gives conditions for deciding whether aninfinite number of rare events will almost surely happen. In this article, wepropose an extension of Borel Cantelli Lemma to characterize the multipleoccurrence of events on the same time scale. Our results imply multipleLogarithm Laws for recurrence and hitting times, as well as Poisson Limit Lawsfor systems which are exponentially mixing of all orders. The applicationsinclude geodesic flows on compact negatively curved manifolds, geodesicexcursions, Diophantine approximations and extreme value theory for dynamicalsystems.

Multiple Borel Cantelli Lemma in dynamics and MultiLog law for recurrence