A Poisson limit theorem for toral automorphisms

Manfred Denker, Mikhail Gordin, Anastasya Sharova

Research output: Contribution to journalArticle

20 Scopus citations

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 ([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 (Gn) n≥0 be a sequence of measurable sets with no periodic points among its accumulation points and such that P(Gn) → 0 as n → ∞, and let (s(n))n>0 be a sequence of positive integers such that limn→∞s(n)P(Gn) = 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=1s(n)1Gn oTi-1 = k) → λk exp (-λ)/k! as n → ∞. Of independent interest is an upper mixing-type estimate, which is one of our main tools.

Original languageEnglish (US)
Pages (from-to)1-20
Number of pages20
JournalIllinois Journal of Mathematics
Volume48
Issue number1
DOIs
StatePublished - Jan 1 2004

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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