Poisson limit for two-dimensional toral automorphisms driven by continued fractions

M. Gordin   ; M. Denker

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 18, Tome 408 (2012), pp. 131-153

Résumé

Generalizing powers of a single hyperbolic automorphism of the two-dimensional torus, we consider some class of sequences of such automorphisms. Technically such sequences are determined by means of continued fraction expansions of a pair of real numbers. As a substitute for the pair of foliations in the classical hyperbolic theory, every sequence of this class has a sequence of asymptotically stable and a sequence of asymptotically unstable foliations. We prove a kind of the Poisson limit theorem for such sequences extending a method used earlier by A. Sharova and the present authors to prove a Poisson limit theorem for powers of a single hyperbolic automorphisms of the torus. Possible generalizations are briefly discussed.

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