On differentiability of SRB states for partially hyperbolic systems

Abstract

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 νe be any u-Gibbs state for f e. We prove (Theorem 1) that for any C ∞ function A the map e→νe(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.