6 Y. KIFER where B : X × M → Rd is Lipschitz in both variables and the maps Fx : M → M are smooth and depend smoothly on the parameter x ∈ Rd. Introducing the map Φε(x, y) = (Xε x,y (1), Y ε x,y (1)) = (x + εB(x, y), Fxy) we can also view this setup as a perturbation of the map Φ(x, y) = (x, Fxy) de- scribing an ideal system where parameters x ∈ Rd do not change. Assuming that Fx, x ∈ Rd are C2 depending on x families of either C2 expanding transformations or C2 Axiom A diffeomorphisms in a neighborhood of an attractor Λx we will derive large deviations estimates for the difference Xε x,y (n) − ¯ ε x (n) where ¯ ε = ¯ ε x solves the equation (1.1.11) d ¯ ε (t) dt = ε ¯( ¯ ε (t)), ¯ ε (0) = x where ¯(x) = B(x, y)dµSRB(y) x and µSRB x is the corresponding SRB invariant measure of Fx on Λx. The discrete time results are obtained, essentially, by simpli- fications of the corresponding arguments in the continuous time case which enable us to describe ”very long” time behavior of the slow motion also in the discrete time case. Since our methods work not only for fast motions being Axiom A dif- feomorphisms but also when they are expanding transformations we can construct simple examples satisfying conditions of our theorems and exhibiting correspond- ing effects. In particular, we produce in Section 1.9 computational examples which demonstrate transitions of the slow motion between neighborhoods of attractors of the averaged system. A series of related results for the case when ordinary differential equations in (1.1.1) are replaced by fully coupled stochastic differential equations appeared in [45], [77]–[79], [66], and [5]. Hasselmann’s nonlinear (strong) diffusion approxima- tion of the slow motion in the fully coupled stochastic differential equations setup was justified in [10]. When the fast process does not depend on the slow one such results were obtained in [44], [29], and [54]. Especially relevant for our results here is [78] and we employ some elements of the probabilistic strategy from this paper. Still, the methods there are quite different from ours and they are based heavily, first, on the Markov property of processes emerging there and, secondly, on uniformity and nondegeneracy of the fast diffusion term assumptions which cannot be satisfied in our circumstances as our deterministic fast motions are very degen- erate from this point of view. Note that the proof in [78] contains a vicious cycle and substantial gaps which recently were essentially fixed in [79]. Some of the dynamical systems technique here resembles [49] but the dependence of the fast motion on the slow one complicates the analysis substantially and requires addi- tional machinery. A series of results on Cramer’s type asymptotics for fully coupled averaging with Axiom A diffeomorphisms as fast motions appeared recently in [4]– [7]. Observe that the methods there do not work for continuous time Axiom A dynamical systems considered here, they cannot lead, in principle, to the standard large deviations estimates of our work and they deal with deviations of Xε from the averaged motion only at the last moment and not of its whole path. Various limit theorems for the difference equations setup (1.1.10) with partially hyperbolic fast motions were obtained recently in [20] and [21]. The study of deviations from the averaged motion in the fully coupled case seems to be quite important for applications, especially, from phenomenological point of view. In addition to perturbations of Hamiltonian systems mentioned above

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