Differentiation of SRB states

Abstract

Let f be a diffeomorphism of a manifold M, and ρf a (generalized) SRB state for f. If suppρf is a hyperbolic compact set we show that the map f → ρf is differentiable in a suitable functional setup, and we compute the derivative. When suppρf is an attractor, the derivative is given by δρf(Φ) = Σ∝n=0ρf〈grad(Φ ○ fn), X〉 where X is the vector field δf ○ f-1. This formula can be extended to time dependent situations and also, at least formally, to nonuniformly hyperbolic situations. The above results will find their use in the study of the Onsager reciprocity relations and the fluctuation-dissipation formula of nonequilibrium statistical mechanics.