Differentiation of SRB states for hyperbolic flows

Abstract

Let the ${\mathcal{C}}^3$ vector field $\mathcal{X}+aX$ on $M$ define a flow $(f_a^t)$ with an Axiom A attractor ${\Lambda }_a$ depending continuously on $a\in (-\epsilon ,\epsilon )$. Let ${\rho }_a$ be the SRB measure on ${\Lambda }_a$ for $(f_a^t)$. If $A\in {\mathcal{C}}^2(M)$, then $a\mapsto {\rho }_a(A)$ is ${\mathcal{C}}^1$ on $(-\epsilon ,\epsilon )$ and $d{\rho }_a(A)/da$ is the limit when $\omega \to 0$ with $\mathrm{Im}\omega >0$ of $${\int }_0^{\infty }{e}^{i\omega t}dt\int {\rho }_a(dx)X(x)\cdot {\nabla }_x(A\circ f_a^t)$$

22 pages