Annales de l'Institut Henri Poincaré / Analyse Non Lineaire 35 (2018) 859-885.
Alexey Korepanov, Zemer Kosloff and Ian Melbourne
Abstract
We prove statistical limit laws for sequences of Birkhoff sums of the type Σ0≤j≤n-1vn o 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 homogenization result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.