Cramér Asymptotics in the Averaging Method for Systems with Fast Hyperbolic Motions

A dynamical system $w'=S(w,z,\varepsilon )$, $z'=z+\varepsilon v(w,z,\varepsilon )$ is considered. It is assumed that slow motions are determined by the vector field $v(w,z,\varepsilon )$ in the Euclidean space and fast motions occur in a neighborhood of a topologically mixing hyperbolic attractor. For the difference between the real and averaged slow motions, the central limit theorem is proved and sharp asymptotics for the probabilities of large deviations (that do not exceed $\varepsilon ^\delta$) are calculated; the exponent $\delta$ depends on the smoothness of the system and approaches zero as the smoothness increases.