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.

@article{AIHPC_2018__35_4_859_0, author = {Korepanov, A. and Kosloff, Z. and Melbourne, I.}, title = {Martingale{\textendash}coboundary decomposition for families of dynamical systems}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {859--885}, publisher = {Elsevier}, volume = {35}, number = {4}, year = {2018}, doi = {10.1016/j.anihpc.2017.08.005}, mrnumber = {3795019}, zbl = {1406.37027}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.08.005/} }

TY - JOUR AU - Korepanov, A. AU - Kosloff, Z. AU - Melbourne, I. TI - Martingale–coboundary decomposition for families of dynamical systems JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 859 EP - 885 VL - 35 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2017.08.005/ DO - 10.1016/j.anihpc.2017.08.005 LA - en ID - AIHPC_2018__35_4_859_0 ER -

%0 Journal Article %A Korepanov, A. %A Kosloff, Z. %A Melbourne, I. %T Martingale–coboundary decomposition for families of dynamical systems %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 859-885 %V 35 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2017.08.005/ %R 10.1016/j.anihpc.2017.08.005 %G en %F AIHPC_2018__35_4_859_0

Korepanov, A.; Kosloff, Z.; Melbourne, I. Martingale–coboundary decomposition for families of dynamical systems. Annales de l'I.H.P. Analyse non linéaire, Volume 35 (2018) no. 4, pp. 859-885. doi : 10.1016/j.anihpc.2017.08.005. http://www.numdam.org/articles/10.1016/j.anihpc.2017.08.005/

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