# Limiting entry times distribution for arbitrary null sets SETS

## TL;DR

This paper develops a method to determine the limiting distribution of return times for arbitrary sets, showing it is generally compound Poisson, and applies it to coupled map lattices, recovering known distributions in special cases.

## Contribution

It introduces a general approach to derive the limiting return times distribution for arbitrary sets, linking it to cluster sizes and extending known results to new contexts.

## Key findings

- Limiting return times distribution is compound Poisson for arbitrary sets.
- Special case of periodic points yields Pólya-Aeppli distribution.
- Application to coupled map lattices confirms theoretical predictions.

## Abstract

We describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound Poisson distributed. We establish a relation between the limiting return times distribution and the probability of the cluster sizes, where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity. In the special case of periodic points we recover the known P\'olya-Aeppli distribution which is associated with geometrically distributed cluster sizes. We apply this method to several examples the most important of which is synchronisation of coupled map lattices. For the invariant absolutely continuous measure we establish that the returns to the diagonal is compound Poisson distributed where the coefficients are given by certain integrals along the diagonal.