4 Y. KIFER setup with the latter fast motions examples of nonconvergence as ε → 0 for large sets of initial conditions (see Remark 1.2.12). In Hamiltonian systems, which are a classical object for applications of aver- aging methods, the whole space is fibered into manifolds of constant energy. For some mechanical systems these manifolds have negative curvature with respect to the natural metric and their motion is described by geodesic flows there. Hyperbolic Hamiltonian systems were discussed, for instance, in [64] and a specific example of a particle in a magnetic field leading to such systems was considered recently in [74]. Of course, these lead to Hamiltonian systems which are far from integrable. Such situations fall in our framework and they are among main motivations for this work. This suggests to consider the equation (1.1.1) on a (locally trivial) fiber bundle M = {(x, y) : x ∈ U, y ∈ Mx} with a base U being an open subset in a Riemannian manifold N and fibers Mx being diffeomorphic compact Riemannian manifolds (see [75]). On the other hand, M has a local product structure and if B is bounded then the slow motion stays in one chart during time intervals of order ∆/ε with ∆ small enough. Hence, studying behavior of solutions of (1.1.1) on each such time interval separately we come back to the product space Rd × M setup and will only have to piece results together to see the picture on a larger time interval of length T/ε. We assume in the first part of this work that b(x, y) is C2 in x and y and that for each x in a closure of a relatively compact domain X the flow Fx t is Anosov or, more generally, Axiom A in a neighborhood of an attractor Λx. Let µx SRB be the Sinai-Ruelle-Bowen (SRB) invariant measure of Fx t on Λx and set ¯(x) = B(x, y)dµx SRB (y). It is known (see [16]) that the vector field ¯(x) is Lipschitz continuous in x, and so the averaged equations (1.1.4) and (1.1.6) have unique solutions ¯ ε (t) and ¯(t) = ¯ ε (t/ε). Still, in general, the measures µx SRB are singular with respect to the Riemannian volume on M, and so the method of [1] cannot be applied here. We proved in [55] that, nevertheless, (1.1.7) still holds true in this case, as well, and, moreover, the measure in (1.1.7) can be estimated by e−c/ε with some c = c(δ) 0. The convergence (1.1.7) itself without an exponential estimate can be proved by another method (see [57]) which can be applied also to some partially hyperbolic fast motions . An extension of the averaging principle in the sense of convergence of Young measures is discussed in Section 1.11. Once the convergence of Zx,y(t) ε = Xx,y(t/ε) ε to ¯ x (t) = ¯ ε x (t/ε) as ε → 0 is established it is interesting to study the asymptotic behavior of the normalized error (1.1.8) V ε,θ x,y (t) = εθ−1(Zε x,y (t) − ¯ x (t)), θ ∈ [ 1 2 , 1]. Namely, in our situation it is natural to study the distributions m{y : V ε,θ x,y (·) ∈ A} as ε → 0 where m is the normalized Riemannian volume on M and A is a Borel subset in the space C0T of continuous paths ϕ(t), t ∈ [0, T ] on Rd. We will obtain in this work large deviations bounds for Vx,y ε = Vx,y ε,1 which will give, in particular, the result from [55] saying that (1.1.9) m{y : Vx,y ε 0,T δ} → 0 as ε → 0 exponentially fast in 1/ε where · 0,T is the uniform norm on C0T . However, the main goal of this work is not to provide another derivation of (1.1.9) but to obtain precise upper and lower large deviations bounds which not only estimate measure

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