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.