On differentiability of SRB states for partially hyperbolic systems

Consider a one parameter family of diffeomorphisms f such that f ₀ is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let be any u-Gibbs state for f . We prove (Theorem 1) that for any C function A the map (A) is differentiable at =0. This implies (Corollary 2.2) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to . 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.