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

Abstract

<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 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 on finite volume hyperbolic manifolds, Diophantine approximations and extreme value theory for dynamical systems.</p>

Poisson Limit Theorems for Systems with Product Structure

“…The reason is that, say if T is Anosov, the condition (BR(x * , y * )) is automatically satisfied for a.e x * and all y * . Due to Remark 4(ii) condition (LR(x * )) is satisfied almost everywhere, furthermore, as shown in [DFL22b], T itself satisfies the PLT almost everywhere, and so in particular (NSR(x * )) holds almost everywhere. So in order to apply Theorem 5 on T × R, for some map R, we just have to check (EE), i.e there are L 2 bounds on equidistribution in R. (EE) can be verified for a big class of maps, in particular it holds if R satisfies a summable decay of correlation.…”