Stability of Statistical Properties in Two-Dimensional Piecewise Hyperbolic Maps

Details Stability of Statistical Properties in Two-Dimensional Piecewise Hyperbolic Maps Stability of Statistical Properties in Two-Dimensional Piecewise Hyperbolic Maps

2006-02-03

arxiv.org/abs/math/0602067v2

Mathematics

Dynamical Systems

35 pages. The proof of Lemma 5.7 has been corrected. To appear in Trans. of Amer. Math. Soc

Scientific paper

We investigate the statistical properties of a piecewise smooth dynamical system by studying directly 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, but we are confident that the present approach can be successful in much greater generality (we hope including higher dimensional billiards). 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.

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