On computing derivatives of transfer operators and linear responses in higher dimensions

We show that the derivative of the transfer operator with respect to perturbations of the map is a divergence. We show that, in high dimensions, approximating singular measures by isotropic finite-elements, and computing the derivative operator on such approximate measures, are both expensive. We show the equivalence between the operator and ensemble version of the linear response formula, and discuss how to combine both formula, as done by the fast linear response algorithm, to reduce cost.