  The work treats systems combining slow and fast motions depending on each
other where fast motions are perturbations of families of either dynamical
systems or Markov processes with freezed slow variable. In the first case we
consider hyperbolic dynamical systems and in the second case we deal with
random evolutions which are combinations of diffusions and continuous time
Markov chains. We study first large deviations of the slow motion from the
averaged one and then use these results together with some Markov property type
arguments in order to describe very long time behavior of the slow motion such
as its transitions between attractors of the averaged system.
