  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.
