Martingale–coboundary decomposition for families of dynamical systems

  • A. Korepanov

    Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
  • Z. Kosloff

    Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
  • I. Melbourne

    Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

Abstract

We prove statistical limit laws for sequences of Birkhoff sums of the type where 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 is replaced by a fixed transformation T, and which is particularly effective in the case when varies with n.

In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family 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.

Cite this article

A. Korepanov, Z. Kosloff, I. Melbourne, Martingale–coboundary decomposition for families of dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 4, pp. 859–885

DOI 10.1016/J.ANIHPC.2017.08.005