Linear response for Dirac observables of Anosov diffeomorphisms

We consider a ${\mathcal{C}}^3$ family $t\mapsto f_t$ of ${\mathcal{C}}^4$ Anosov diffeomorphisms on a compact Riemannian manifold $M$. Denoting by ${\rho }_t$ the SRB measure of $f_t$, we prove that the map $t\mapsto \int \theta d{\rho }_t$ is differentiable if $\theta$ is of the form $\theta (x)=h(x)\delta (g(x)-a)$, with $\delta$ the Dirac distribution, $g:M\to \mathbb{R}$ a ${\mathcal{C}}^4$ function, $h:M\to \mathbb{R}$ a ${\mathcal{C}}^3$ function and $a$ a regular value of $g$. We also require a transversality condition, namely that the intersection of the support of $h$ with the level set $\{g(x)=a\}$ is foliated by ‘admissible stable leaves’.