Martingale–coboundary decomposition for families of dynamical systems

We prove statistical limit laws for sequences of Birkhoff sums of the type ∑_j=0 ⁿ⁻¹v_n∘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.