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Consider a one parameter family of diffeomorphisms f e such that f 0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let ? e be any u-Gibbs state for f e. We prove (Theorem 1) that for any C 8 function A the map e?? e( A) is differentiable at e=0. This implies (Corollary 2.2) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to e. We apply this result (Corollary 3.3) to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique SRB measure with good statistical properties.

On differentiability of SRB states for partially hyperbolic systems