Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables

Abstract

We consider a smooth one-parameter family $t \to f_t$ of diffeomorphisms with compact transitive Axiom A attractors. Our first result (corrected) is that for any function $G$ in the Sobolev space $H^r_p$, with $p>1$ and $0<r<1/p$, the map $R(t)$ sending $t$ to the average of $G$ with respect to the SRB measure of $f_t$ is $\alpha$-H\"older continuous for all $\alpha <r- |log \mathcal J|/(p|log \nu_s|$) where $\mathcal J\le 1$ is the strongest volume contraction and $\nu_s<1$ is the weakest contraction. This applies to $\theta(x)=h(x)\Theta(g(x)-a)$ (for all $\alpha <1- |log \mathcal J|/|log \nu_s|$) for $h$ and $g$ smooth and $\Theta$ the Heaviside function, if $a$ is not a critical value of $g$. Our second result says that for any such function so that, in addition, the intersection of the set of points $x$ so that $g(x)=a$ with the support of $h$ is foliated by "admissible stable leaves" of $f_t$, the map $R(t)$ is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.) Obtaining linear response or fractional response for such observables is motivated by extreme-value theory. --- Second version, following the referee's comments: We explain better the cone choices around (2.4). Appendix A contains information on the Banach spaces. We added the paragraph containing (2.6) in the proof of Theorem 2.1. In the proof of Theorem 3.3, we do not need to introduce mollifiers. However, the new argument around (2.6) is not available here, so we must replace the pair $(u-1, |s-1|)$ by $(u-2, |s-2|)$. This is why we now assume that $h$ is $C^3$ and that $g$ and the foliations are $C^4$. --- Third version: We have added a corrigendum modifying the first result (Theorem 2.1).