Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions

In the study of systems which combine slow and fast motions which depend on each other (fully coupled setup) whenever the averaging principle can be justified this usually can be done only in the sense of $L^1$-convergence on the space of initial conditions. When fast motions are hyperbolic (Axiom A) flows or diffeomorphisms (as well as expanding endomorphisms) for each freezed slow variable this form of the averaging principle was derived in [19] and [20] relying on some large deviations arguments which can be applied only in the Axiom A or uniformly expanding case. Here we give another proof which seems to work in a more general framework, in particular, when fast motions are some partially hyperbolic or some nonuniformly hyperbolic dynamical systems or nonuniformly expanding endomorphisms.