THE AVERAGING METHOD FOR PERTURBATIONS OF MIXING FLOWS

For a class of multiphase averaging problems in which the unperturbed flow of the fast variables is a transitive Anosov flow or is 'sufficie ntly rapidly mixing', we obtain what we believe is an optimal power-la w estimate, in the small parameter, of the rate at which the average m aximum difference between solutions of the exact and averaged problems converges to zero. This verifies and extends an earlier conjectural r emark by V. I. Arnold.