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.