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.
Transactions of the American Mathematical Society
Demers, Mark and Liverani, Carlangelo, "Stability of Statistical Properties in Two-Dimensional Piecewise Hyperbolic Maps" (2008). Mathematics Faculty Publications. 2.
M. F. Demers, and C. Liberani. Stability of Statistical Properties in Two-Dimensional Piecewise Hyperbolic Maps, Transactions of the American Mathematical Society. 360(9), 4777-4814.