A Rigorous Computational Approach to Linear Response

Abstract

We present a general setting in which the formula describing the linear response of the physical measure of a perturbed system can be obtained. In this general setting we obtain an algorithm to rigorously compute the linear response. We apply our results to expanding circle maps. In particular, we present examples where we compute, up to a pre-specified error in the $L^{\infty}$-norm, the response of expanding circle maps under stochastic and deterministic perturbations. Moreover, we present an example where we compute, up to a pre-specified error in the $L^1$-norm, the response of the intermittent family at the boundary; i.e., when the unperturbed system is the doubling map.Comment: Revised version following reports. A new example which contains the computation of the linear response at the boundary of the intermittent family has been adde