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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