Statistical properties of mostly contracting fast-slow partially hyperbolic systems

We consider a class of C 4 partially hyperbolic systems on T 2 described by maps Fε(x, θ) = (f(x, θ), θ + εω(x, θ)) where f(·, θ) are expanding maps of the circle. For sufficiently small ε and ω generic in an open set, we precisely classify the SRB measures for Fε and their statistical properties, including exponential decay of correlation for H¨older observables with explicit and nearly optimal bounds on the decay rate.