  A general approach to a broad class of asymptotic problems related to
long-time influence of small perturbations, of both deterministic and
stochastic type, is presented in the paper. The main characteristic of this
influence is a limiting motion on the simplex of invariant probability measures
of the non-perturbed system in an appropriate time scale. The main tools we use
in the paper are limit theorems for large deviations, modified averaging
principle, and diffusion approximation.
