Multiple Borel Cantelli Lemma in dynamics and MultiLog law for recurrence

TL;DR

This paper extends the Borel Cantelli Lemma to multiple events in dynamical systems, deriving new laws for recurrence, hitting times, and Poisson limits with applications in geometry, number theory, and extreme value analysis.

Contribution

It introduces a multiple Borel Cantelli Lemma for dynamical systems, enabling analysis of simultaneous event occurrences and deriving new logarithm and Poisson limit laws.

Findings

Multiple Logarithm Laws for recurrence and hitting times.

Poisson Limit Laws for exponentially mixing systems.

Applications to geodesic flows, Diophantine approximation, and extreme value theory.

Abstract

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.