Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence

<p style='text-indent:20px;'>A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose extension to characterize the multiple occurrence on same time scale. Our results imply Logarithm Laws recurrence and hitting times, as well Poisson Limit systems which are exponentially mixing all orders. The applications include geodesic flows compact negatively curved manifolds, excursions finite volume hyperbolic Diophantine approximations extreme value theory dynamical systems.</p>