### Abstract

We introduce a new method of proving Poisson limit laws in the theory of dynamical systems, which is based on the Chen-Stein method ([8], [21]) combined with the analysis of the homoclinic Laplace operator in [12] and some other homoclinic considerations. This is accomplished for the hyperbolic toral automorphism T and the normalized Haar measure P. Let (G_{n}) _{n≥0} be a sequence of measurable sets with no periodic points among its accumulation points and such that P(G_{n}) → 0 as n → ∞, and let (s(n))_{n>0} be a sequence of positive integers such that lim_{n→∞}s(n)P(G_{n}) = λ for some λ > 0. Then, under some additional assumptions about (G _{n})_{n≥0}, we prove that for every integer k ≥ 0 P(∑_{i=1}^{s(n)}1G_{n} oT^{i-1} = k) → λ^{k} exp (-λ)/k! as n → ∞. Of independent interest is an upper mixing-type estimate, which is one of our main tools.

Original language | English (US) |
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Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Illinois Journal of Mathematics |

Volume | 48 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2004 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Illinois Journal of Mathematics*,

*48*(1), 1-20. https://doi.org/10.1215/ijm/1258136170