A Poisson limit theorem for toral automorphisms

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.