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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 19/2002

Manfred DENKER, Mikhail GORDIN, and Anastasya SHAROVA
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A Poisson limit theorem for toral automorphisms

This preprint was accepted November 4, 2002

Contact:
` M. Gordin`

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{7, 20})
combined with the
analysis of the homoclinic Laplace operator in \cite{11} and some other
homoclinic considerations. This is
accomplished for
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 $k \ge 0$
$$P(\sum_{i=1}^{s(n)} 1_{G_n}\circ T^{i-1} = k) \to \lambda^k \exp { (- \lambda)}
/k! $$ as $n \to \infty$
Of independent iterest is an upper mixing-type estimate which is one
of our main tools.

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