Linear response for Dirac observables of Anosov diffeomorphisms

Abstract

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, $g:M\rightarrow \mathbb{R}$ a $\mathcal{C}^4$ function, $h:M\rightarrow\mathbb{R}$ 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'.