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

We consider a smooth one-parameter family t\mapsto ≤ft( {{f}_t}:M\to M\right) of diffeomorphisms with compact transitive Axiom A attractors {{ Λ }_t} , denoting by \text{d}{ρ_t} the SRB measure of {{f}_t}{{|}_{{{ Λ t}}}} . Our first result is that for any function θ in the Sobolev space H_p^r(M) , with 1 and 0  <  r  <  1/p, the map t\mapsto {\int}θ \text{d}{ρ_t} is α-Hölder continuous for all α . This applies to θ (x)=h(x) \Theta ≤ft(g(x)-a\right) (for all α <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 θ (x)=h(x) \Theta ≤ft(g(x)-a\right) so that in addition the intersection of ≤ft\{x|g(x)=a\right\} with the support of h is foliated by ‘admissible stable leaves’ of f _t , the map t\mapsto {\int}θ \text{d}{ρ_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.