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

摘要

We consider a smooth one-parameter family t bar right arrow (f(t) : M - M) of diffeomorphisms with compact transitive Axiom A attractors Lambda(t), denoting by d rho(t) the SRB measure of f(t)vertical bar Lambda(t). Our first result is that for any function theta in the Sobolev space H-p(r) (M), with 1 p infinity and 0 r 1/p, the map t bar right arrow integral theta d rho(t) is alpha-Holder 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 vertical bar g(x) = a} with the support of h is foliated by 'admissible stable leaves' of ft, the map t bar right arrow integral 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.