Differentiation of SRB states for hyperbolic flows

TL;DR

This paper investigates how SRB measures for hyperbolic flows depend smoothly on parameters, providing a differentiability result and an explicit formula for the derivative of the measure with respect to the parameter.

Contribution

It establishes the ${ m C}^1$ dependence of SRB measures on parameters for hyperbolic flows and derives an explicit formula for the derivative.

Findings

SRB measures depend ${ m C}^1$ on parameters in hyperbolic flows

Provides an explicit formula for the derivative of SRB measures

Connects the derivative to a limit involving a complex integral

Abstract

Let the ${\cal C}^3$ vector field ${\cal X}+aX$ on $M$ define a flow $(f^t_a)$ 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 $(f^t_a)$. If $A\in{\cal C}^2(M)$, then $a\mapsto\rho_a(A)$ is ${\cal C}^1$ on $(-\epsilon,\epsilon)$ and $d\rho_a(A)/da$ is the limit when $\omega\to0$ with ${\rm Im}\omega>0$ of $$ \int_0^\infty e^{i\omega t}dt \int\rho_a(dx) X(x)\cdot\nabla_x(A\circ f_a^t) $$