## Markov approximations and decay of correlations for Anosov flows.(English)Zbl 0911.58028

Suppose $$\phi^t$$ is a $$C^2$$ Anosov flow on a compact Riemannian manifold of dimension 3. Suppose $$\phi^t$$ is topologically mixing, then its stable and unstable foliations are ‘nonintegrable’. Assume in addition that this nonintegrability is ‘uniform’. Denote by $$\mu$$ the Sinai-Bowen-Ruelle measure of $$\phi^t$$. It is shown that then there are constants $$c$$ and $$a$$ such that for any generalized Hölder continuous functions $$F$$ and $$G$$ there exists a constant $$v= v(F,G)$$ such that the time correlation $$C_{F,G}(t)$$, given by $C_{F,G}(t)= \int F(\phi^t(x)) G(x)d\mu(x)- (\int G d\mu)\cdot (\int F d\mu),$ satisfies $$| C_{F,G}(t)|\leq v\cdot(c\exp(- a\sqrt t))$$. The dependence of $$c$$ and $$a$$ on $$\phi^t$$ is continuous in $$C^1$$.
Reviewer: H.Crauel (Berlin)

### MSC:

 37D99 Dynamical systems with hyperbolic behavior
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