09-144 Miaohua Jiang

Exact Derivative formula of the potential function of the SRB measure (317K, pdf) Aug 24, 09

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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.

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