Miaohua Jiang
Exact Derivative formula of the potential function of the SRB measure
(317K, pdf)
ABSTRACT. For a $C^r$-diffeomorphism ($r \ge 3$) $f$ on a smooth compact Riemannian manifold
possessing a hyperbolic attractor, the potential function for the
SRB measure $-\log J^uf(h_f(x))$ is differentiable with respect to $f$ in a
$C^r$-neighborhood of $f$. We show that if we calculate the unstable Jacobian
$J^u f$ with respect to a H\"older continuous metric $\omega_0$ under which the stable and unstable subspaces are orthogonal,
the derivative formula in a given direction $\delta f,$ a vector field on $M$ evaluated at $f(x)$, is
given exactly by
\[ \delta ( \log J^u_{0} f (h_f(x))) = \Div^u_\rho X^u(f(x)) \]
where $X^u, X^s $
are the projections of the vector field $ \delta f\circ f^{-1}$
onto unstable and stable subbundles, $\Div^u_\rho X^u $ is the divergence of
$X^u$ with respect to the volume form induced by the SRB measure $\rho$ of $f$, and $J^u_{0} f$ is the unstable Jacobian
with respect to the metric $\omega_0$ on the unstable
manifold of $f$. This result complements Ruelle's formula by identifying a metric under which the
coboundary term can be determined exactly and also gives an alternative proof of the derivative
formula of the SRB measure.