Linear response for Dirac observables of Anosov diffeomorphisms

TL;DR

This paper proves differentiability of SRB measures for a family of Anosov diffeomorphisms when observables are Dirac distributions supported on regular level sets, under transversality conditions.

Contribution

It establishes linear response formulas for Dirac observables of Anosov systems, extending previous results to singular measures supported on level sets.

Findings

Differentiability of SRB integrals for Dirac observables under transversality.

Explicit linear response formula for Dirac observables.

Conditions ensuring regularity of the response in hyperbolic dynamics.

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'.