# Rare event process and entry times distribution for arbitrary null sets on compact manifolds

## TL;DR

This paper links rare event processes and entry times distributions for measure-zero sets on compact manifolds, showing convergence to compound Poisson distributions in certain dynamical systems, with explicit parameter formulas and criteria for Poisson limits.

## Contribution

It establishes a general equivalence between rare event processes and entry times distributions for null sets, and demonstrates convergence to compound Poisson distributions in specific dynamical systems.

## Key findings

- Rare event process and entry times distribution are equivalent for null sets.
- Convergence to compound Poisson distributions in differentiable maps modeled by Young's towers.
- Explicit formulas for distribution parameters and criteria for Poisson limits.

## Abstract

We establish the general equivalence between rare event process for arbitrary continuous functions whose maximal values are achieved on non-trivial sets, and the entry times distribution for arbitrary measure zero sets. We then use it to show that the for differentiable maps on a compact Riemannian manifold that can be modeled by Young's towers, the rare event process and the limiting entry times distribution both converge to compound Poisson distributions. A similar result is also obtained on Gibbs-Markov systems, for both cylinders and open sets. We also give explicit expressions for the parameters of the limiting distribution, and a simple criterion for the limiting distribution to be Poisson. This can be applied to a large family of continuous observables that achieve their maximum on a non-trivial set with zero measure.