Quantitative statistical properties of two-dimensional partially hyperbolic systems

We study a class of two-dimensional partially hyperbolic systems, not necessarily skew products, in an attempt to develop a general theory. As a main result, we provide explicit conditions for the existence of finitely many physical measures (and SRB) and prove exponential decay of correlations for mixing measures. In addition, we obtain precise information on the regularity of such measures (they are absolutely continuous with respect to Lebesgue with density in some Sobolev space). To illustrate the scope of the theory, we show that our results apply to the case of fast-slow partially hyperbolic systems, and for such systems we obtain more precise results on the structure of the SRB measures.