Optimal linear response for Anosov diffeomorphisms

Abstract

It is well known that an Anosov diffeomorphism $T$ enjoys linear response of its SRB measure with respect to infinitesimal perturbations $\dot{T}$. For a fixed observation function $c$, we develop a theory to optimise the response of the SRB-expectation of $c$. Our approach is based on the response of the transfer operator on the anisotropic Banach spaces of Gou\"ezel--Liverani. We prove that the optimising perturbation $\dot{T}$ is unique for non-degenerate response functions and provide explicit expressions for the Fourier coefficients of $\dot{T}$. We develop an efficient Fourier-based numerical scheme to approximate the optimal vector field $\dot{T}$, along with a proof of convergence. The utility of our approach is illustrated in two numerical examples, by localising SRB measures with small, optimally selected, perturbations.