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Averaging principle for fully coupled dynamical systems and large deviations

In the study of systems which combine slow and fast motions, the averaging principle suggests that a good approximation of the slow motion on long time intervals can be obtained by averaging its parameters over the fast variables. When the slow and fast motions depend on each other (fully coupled), as is usually the case, for instance, in perturbations of Hamiltonian systems, the averaging prescription cannot always be applied, and when it does work, this is usually only in some averaged with respect to initial conditions sense. In this paper we first give necessary and sufficient conditions for the averaging principle to hold (in the above sense) and then, relying on some large deviations arguments, verify them in the case when the fast motions are hyperbolic (Axiom A) flows for each freezed slow variable. It turns out that in this situation the Lebesgue measure of initial conditions with bad averaging approximation tends to zero exponentially fast as the parameter tends to zero.