Abstract/Summary

We consider a smooth one-parameter family $t\mapsto (f_t:M\to M)$ of diffeomorphisms with compact transitive Axiom A attractors $\Lambda_t$, denoting by $d \rho_t$ the SRB measure of $f_t|_{\Lambda_t}$. Our first result is that for any function $\theta$ in the Sobolev space $H^r_p(M)$, with $1<p<\infty$ and $0<r<1/p$, the map $t\mapsto \int \theta\, d\rho_t$ is $\alpha$-H\"older continuous for all $\alpha <r$. This applies to $\theta(x)=h(x)\Theta(g(x)-a)$ (for all $\alpha <1$) 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 $\theta(x)=h(x)\Theta(g(x)-a)$ so that in addition the intersection of $\{ x\mid g(x)=a\}$ with the support of $h$ is foliated by ``admissible stable leaves'' of $f_t$, the map $t\mapsto \int \theta\, d\rho_t$ is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.) Obtaining linear response or fractional response for such observables $\theta$ is motivated by extreme-value theory.