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