Exponential mixing for singular skew-products

We study skew-products of the form $(x,u)\mapsto (fx,u+\phi (x))$ where $f$ is a non-uniformly expanding map on a manifold $X$ and $\phi :X\to {\mathbb{S}}^1$ is piecewise ${\mathcal{C}}^1$. If the systems satisfies mild assumptions (in particular singular behaviour of $\phi$ is permitted) then we prove that the map mixes exponentially with respect to the unique SRB measure. This extends previous results by allowing singular behaviour in the fibre map.