96-499 David Ruelle

Differentiation of SRB states (43K, TeX) Oct 21, 96

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Abstract. Let $f$ be a diffeomorphism of a manifold $M$, and $\rho_f$ a (generalized) SRB state for $f$. If ${\rm supp}\rho_f$ is a hyperbolic compact set we show that the map $f\mapsto\rho_f$ is differentiable in a suitable functional setup, and we compute the derivative. When ${\rm supp}\rho_f$ is an attractor, the derivative is given by $$ \delta\rho_f(\Phi)=\sum_{n=0}^\infty\rho_f \langle{\rm grad}(\Phi\circ f^n),X\rangle $$ where $X$ is the vector field $\delta f\circ f^{-1}$. This formula can be extended, at least formally, to time dependent situations, and also 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.

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