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

Viviane Baladi, Tobias Kuna, Valerio Lucarini

Published 2016-03-31Version 1

We consider a smooth one-parameter family t -> f_t of diffeomorphisms with compact transitive Axiom A attractors. Our first result is that for any function G in the Sobolev space H^r_p, with p>1 and 0<r<1/p, the map R(t) sending t to the average of G with respect to the SRB measure of f_t is s-H\"older continuous for all s <r. This applies to h(x)H(g(x)-a) (for all s<1) for h and g smooth and H the Heaviside function, if a is not a critical value of g. Our second result says that for any such function so that, in addition, the intersection of the set of points x so that g(x)=a with the support of h is foliated by "admissible stable leaves" of f_t, the map R(t) is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.) Obtaining linear response or fractional response for such observables is motivated by extreme-value theory.