Stability of statistical properties in two-dimensional piecewise hyperbolic maps

We investigate the statistical properties of a piecewise smooth dynamical system by directly studying the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of two-dimensional maps with uniformly bounded second derivative. For the class of systems at hand, we obtain a complete description of the SRB measures, their statistical properties and their stability with respect to many types of perturbations, including deterministic and random perturbations and holes.