2022, 18: 209-289. doi: 10.3934/jmd.2022009

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

1. 

Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, MD 20742-4015, USA

2. 

Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

Received  January 15, 2021 Revised  October 09, 2021

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.

Citation: Dmitry Dolgopyat, Bassam Fayad, Sixu Liu. Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence. Journal of Modern Dynamics, 2022, 18: 209-289. doi: 10.3934/jmd.2022009
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Figure 1.  Proof of Lemma 8.6
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