Martingale-coboundary decomposition for families of dynamical systems

We prove statistical limit laws for sequences of Birkhoff sums of the type Sigma(n-1)(j=0)v(n) circle T-n(j) where T-n is a family of nonuniformly hyperbolic transformations. The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family T-n is replaced by a fixed transformation T, and which is particularly effective in the case when T-n varies with n. In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family T-n consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally forced dispersing billiards. As an application, we prove a homogenisation result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations. (C) 2017 Elsevier Masson SAS. All rights reserved.