Differentiation of SRB States

Dedicated to Klaus Hepp and Walter Hunziker for their 60th birthdays.

Abstract:Let f be a diffeomorphism of a manifold M, and ρ_f a (generalized) SRB state for f. If supp ρ_f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup, and we compute the derivative. When supp ρ_f is an attractor, the derivative is given by

where X is the vector field . 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.