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. To model non-uniformly hyperbolic systems, 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 an exponential mixing property of the SRB measure when conditioned on the map's singularity set. This property appears to be true and is of independent interest. Further study of this conditional mixing property may yield a better picture of linear response theory.