Martingale-coboundary decomposition for families of dynamical systems

Abstract

We prove statistical limit laws for sequences of Birkhoff sums of the type ∑j=0n-1vn(ring operator)Tnj where Tn 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 Tn is replaced by a fixed transformation T, and which is particularly effective in the case when Tn varies with n.In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family Tn 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.