Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems

Abstract

We consider families of fast-slow skew product maps of the form xn+1 = xn + ǫa(xn, yn, ǫ), yn+1 = Tǫyn, where Tǫ is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables x as ǫ → 0. Similar results are obtained also for continuous time systems x˙ = ǫa(x, y, ǫ), y˙ = gǫ(y). Our results include cases where the family of fast dynamical systems consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.