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

We consider a smooth one-parameter family t → ( ft : M → M) of diffeomorphisms with compact transitive Axiom A attractors λt denoting by dpt the SRB measure of fttλt. Our first result is that for any function θ in the Sobolev space Hrp(M), with 1π-rfpagπ ∞ and 0 π r π 1/p, the map tx→ ∫ θ dpt is ?-Hölder continuous for all r. This applies to(x) = h(x) θ (g(x) ? a) (for all >1) for h and g smooth and θ the Heaviside function, if a is not a critical value of g. Our second result says that for any such function -(x) = h(x) θ (g(x) ? a) so that in addition the intersection of {x|g(x) = a} with the support of h is foliated by admissible stable leaves of ft, the map t d-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.