Steklov Institute of Mathematics at St.Petersburg

PREPRINT 19/2002

Manfred DENKER, Mikhail GORDIN, and Anastasya SHAROVA

A Poisson limit theorem for toral automorphisms

This preprint was accepted November 4, 2002
Contact: M. Gordin

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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