Preface
This work studies the long time behavior of slow motions in two scale fully
coupled systems and it consists of two, essentially, independent parts which even
have their own introductions. The first part is written having in mind readers with
strong backgrounds in smooth dynamical systems and it deals with the case of
Axiom A flows as fast motions. The second part is written for probabilists and it
studies the case where fast motions are certain Markov processes such as random
evolutions and, in particular, diffusions. As we noticed already in [47] principal
large deviations results for Axiom A systems and Markov processes (satisfying,
say, the Doeblin condition) follow from a similar scope of ideas and basic theorems
though they rely on quite different machineries and backgrounds. Rate functionals
of large deviations turn out to be Legendre transforms of corresponding topolog-
ical pressures in the dynamical systems case while in the diffusion case they are
obtained in the same way from principal eigenvalues of the corresponding infinitesi-
mal generators. This intrinsic connection is further amplified by the fact that in the
random diffusion perturbations of dynamical systems setup these principal eigen-
values converge to topological pressures when the perturbation parameter tends to
zero (see [46]).
Usually, Markov processes are easier to deal with since we can use the Markov
property there for free while in the dynamical systems case we have to look for some
substitute. We felt that the first part of this work would be quite difficult to follow
for most of probabilists in view of its heavy dynamical systems machinery. By this
reason the second part is written in the way that it can be read independently of
the first one and it relies only on the standard probabilistic background though
the strategies of the proof in both parts are similar with the Markov property
making arguments easier in the second part which also does not require to deal
with geometric pecularities of the hyperbolic deterministic dynamics of the first
part. In order to ensure a convenient independent reading of the second part we
give full arguments there except for very few references to some general proofs in
the first part which do not rely on the specific dynamical systems setup there. Still,
the readers having sufficient background both in dynamical systems and Markov
processes will certainly benefit from having proofs for both cases in one place and
such exposition demonstrates boldly unifying features of these two quite different
objects. We observe, that it could be possible to start with some very general
(though quite unwieldy) assumptions which would enable us to prove similar results
and then verify these assumptions for both cases we are dealing with but we believe
that such exposition would make the paper quite difficult to read for both groups
of mathematicians this work is addressed to.
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