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

Studia Math. 238 (2017) 59-89.

Alexey Korepanov, Zemer Kosloff and Ian Melbourne

Abstract We consider families of fast-slow skew product maps of the form

x_n+1 = x_n + ε a(x_n,y_n,ε), y_n+1 = T_ε y_n,

where T_ε is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables x as ε→0. Similar results are obtained also for continuous time systems

x' = ε a(x,y,ε), y' = g_ε(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.