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

We prove the almost sure invariance principle with rate $o(n^{\epsilon })$ for every $\epsilon >0$ for H"older 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 H'enon maps. The best previously proved rate is $O(n^{1\mathrm{/}4}(\log n{)}^{1\mathrm{/}2}(\log \log n{)}^{1\mathrm{/}4})$. 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.