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arXiv · 1608.01853

Martingale-coboundary decomposition for families of dynamical systems

A. Korepanov·Z. Kosloff·I. Melbourne
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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.

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A. Korepanov, Z. Kosloff, I. Melbourne. 2016-08-05. Martingale-coboundary decomposition for families of dynamical systems. https://arxiv.org/abs/1608.01853

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