A Poisson limit theorem for toral automorphisms

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 (\cite{Ch}, \cite{St}) combined with the analysis of the homoclinic Laplace operator in \cite{Go} and some other homoclinic considerations. This is accomplished for the hyperbolic toral automorphism $T$ and the normalized Haar measure $P$. Let $(G_n)_{n \ge 0}$ be a sequence of measurable sets with no periodic points among its accumulation points and such that $P(G_n) \to 0$ as $n \to \infty$, and let $(s(n))_{n > 0}$ be a sequence of positive integers such that $\lim_{n\to \infty} s(n)P(G_n)=\lambda$ for some $\lambda>0$. Then, under some additional assumptions about $(G_n)_{n \ge 0}$, we prove that for every integer $k \ge 0$

\[ P\left(\sum_{i=1}^{s(n)} \one _{G_n}\circ T^{i-1} = k\right) \to \lambda^k \exp { (- \lambda)} /k! \]

as $n \to \infty$. Of independent interest is an upper mixing-type estimate, which is one of our main tools.