Rates in almost sure invariance principle for dynamical systems with some hyperbolicity

Abstract

We prove the almost sure invariance principle with rate $o(n^{\epsilon })$ for every $\epsilon >0$ 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.

Changed presentation and simplified assumptions. To appear in Communications in Mathematical Physics

References in corpus (4)

• Komlós-Major-Tusnády approximation under dependence
• Strong invariance principles with rate for "reverse" martingales and applications
• Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle
• Explicit coupling argument for nonuniformly hyperbolic transformations

Cited by in corpus (2)

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