Statistical properties of mostly contracting fast-slow partially hyperbolic systems

We consider a class of ${\mathcal{C}}^4$ partially hyperbolic systems on ${\mathbb{T}}^2$ described by maps $F_{\epsilon }(x,\theta )=(f(x,\theta ),\theta +\epsilon \omega (x,\theta ))$ where $f(\cdot ,\theta )$ are expanding maps of the circle. For sufficiently small $\epsilon$ and $\omega$ generic in an open set, we precisely classify the SRB measures for $F_{\epsilon }$ and their statistical properties, including exponential decay of correlation for H"older observables with explicit and nearly optimal bounds on the decay rate.