Martingale–coboundary decomposition for families of dynamical systems

Korepanov, A.; Kosloff, Z.; Melbourne, I.

doi:10.1016/j.anihpc.2017.08.005

Korepanov, A. ; Kosloff, Z. ; Melbourne, I.

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 859-885

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Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2017.08.005

Keywords: Martingale–coboundary decomposition, Nonuniform hyperbolicity, Statistical limit laws, Homogenisation, Fast–slow systems

@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://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.08.005/}
}

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

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