The compound Poisson distribution and return times in dynamical systems

TL;DR

This paper investigates the distribution of return times in dynamical systems, demonstrating that at periodic points, the distribution converges to a compound Poisson distribution, with general theorems and error estimates provided.

Contribution

It establishes the limiting return time distribution as a compound Poisson distribution at periodic points and introduces a general theorem applicable to various settings.

Findings

Return times at periodic points follow a compound Poisson distribution

Error bounds for convergence to the limiting distribution are derived

A general theorem for establishing compound Poisson distributions is presented

Abstract

Previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere Poissonian. Here we study the behaviour of return times at periodic points and show that the limiting distribution is a compound Poissonian distribution. We also derive error terms for the convergence to the limiting distribution. We also prove a very general theorem that can be used to establish compound Poisson distributions in many other settings.