Martingale–coboundary decomposition for families of dynamical systems

Abstract

We prove statistical limit laws for sequences of Birkhoff sums of the type \sum _{j = 0}^{n−1}v_{n} \circ 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.