Poison Limit for Two-Dimensional Toral Automorphism Driven by Continued Fractions

Abstract

Generalizing powers of a single hyperbolic automorphism of the two-dimensional torus, we consider some class sequences of such automorphism. As a substitute for the pair of foliations in the classical hyperbolic theory, every sequence of this class has a stable family of 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 the Poisson limit theorem for powers of a single hyperbolic automorphism of the torus. Possible generalizations are briefly discussed.