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 =
B(x, y)dµx SRB(y). It is known (see [16]) that the vector field
¯(x)
B is Lipschitz
continuous in x, and so the averaged equations (1.1.4) and (1.1.6) have unique
solutions
¯
X
ε(t)
and
¯(t)
Z =
¯
X
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
ε
(t) = Xx,y(t/ε)
ε
to
¯
Z x(t) =
¯
X
ε(t/ε)
x
as ε 0
is established it is interesting to study the asymptotic behavior of the normalized
error
(1.1.8) Vx,y
ε,θ(t)
=
εθ−1(Zx,y ε
(t)
¯
Z
x
(t)), θ [
1
2
, 1].
Namely, in our situation it is natural to study the distributions m{y : Vx,y
ε,θ(·)
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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