On convergence of linear response formulae in some piecewise hyperbolic maps

Abstract When high-dimensional non-uniformly hyperbolic chaotic systems undergo dynamical perturbations, their long-time statistics are generally observed to respond differentiably with respect to the perturbation. Although important in applications, this differentiability, which is thought to be connected to the dimensionality of the system, has remained resistant to rigorous study outside of the one-dimensional setting. To model non-uniformly hyperbolic systems in multiple dimensions, we consider a family of the mathematically tractable class of piecewise smooth hyperbolic maps, the Lozi maps. For these maps, we prove that the existence of a formal derivative of the response reduces to conditional mixing of the SRB measure on the singularity set. Heuristically, this suggests that Lozi maps, and piecewise uniformly expanding maps more generally, should have linear response.