A Poisson limit theorem for Gibbs-Markov maps

TL;DR

This paper proves that for Gibbs-Markov maps, the number of visits to shrinking sets follows a Poisson distribution in the limit, extending classical results to a broader class of dynamical systems.

Contribution

It establishes a Poisson limit theorem for Gibbs-Markov maps, generalizing Doeblin's classical result to these systems.

Findings

Visits to shrinking sets converge to a Poisson distribution

Extension of Doeblin's Poisson limit theorem to continued fractions

Provides a new probabilistic understanding of Gibbs-Markov dynamics

Abstract

We prove for Gibbs-Markov maps that the number of visits to a sequence of shrinking sets with bounded cylindrical lengths converges in distribution to a Poisson law. Applying to continued fractions, this result extends Doeblin's Poisson limit theorem.