  We consider the skew product $F: (x,u) \mapsto (f(x), u + \tau(x))$, where
the base map $f : \mathbb{T}^{1} \to \mathbb{T}^{1}$ is piecewise
$\mathcal{C}^{2}$, covering and uniformly expanding, and the fibre map $\tau :
\mathbb{T}^{1} \to \mathbb{R}$ is piecewise $\mathcal{C}^{2}$. We show the
dichotomy that either this system mixes exponentially or $\tau$ is cohomologous
(via a Lipschitz function) to a piecewise constant.
