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 homogenization result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.

Minor changes. To appear in Annales l'Institut H. Poincare. Anal. Non Lineaire

References in corpus (4)

• Linear response for intermittent maps
• Statistical stability of equilibrium states for interval maps
• Strong invariance principles with rate for "reverse" martingales and applications
• Spatial structure of Sinai-Ruelle-Bowen measures

Cited by in corpus (1)

• Almost Sure Invariance Principle for non-autonomous holomorphic dynamics in ${\mathbb{P}}^k$