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

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