Abstract
We prove the almost sure invariance principle with rate \({o(n^{\varepsilon})}\) for every \({\varepsilon > 0}\) for Hölder continuous observables on nonuniformly expanding and nonuniformly hyperbolic transformations with exponential tails. Examples include Gibbs–Markov maps with big images, Axiom A diffeomorphisms, dispersing billiards and a class of logistic and Hénon maps. The best previously proved rate is \({O(n^{1/4} (\log n)^{1/2} (\log \log n)^{1/4})}\). As a part of our method, we show that nonuniformly expanding transformations are factors of Markov shifts with simple structure and natural metric (similar to the classical Young towers). The factor map is Lipschitz continuous and probability measure preserving. For this we do not require the exponential tails.
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Acknowledgements
This research was supported in part by a European Advanced Grant StochExtHomog (ERC AdG 320977) at the University ofWarwick and an Engineering and Physical Sciences Research Council grant EP/P034489/1 at the University of Exeter. The author is grateful to Mark Holland and Ian Melbourne for support. The author has been very lucky with the referees who were aware of the history of the problem and made invaluable suggestions for improvement of the manuscript. The author is thankful to Christophe Cuny, Jérôme Dedecker and Florence Merlevède for helpful comments and for pointing out a mistake in the construction of semiconjugacy.
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Korepanov, A. Rates in Almost Sure Invariance Principle for Dynamical Systems with Some Hyperbolicity. Commun. Math. Phys. 363, 173–190 (2018). https://doi.org/10.1007/s00220-018-3234-5
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DOI: https://doi.org/10.1007/s00220-018-3234-5