David Ruelle
Differentiation of SRB states
(43K, TeX)
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.