  Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine
increasing map which is injective but not surjective. Such a map $f$ has a
rotation number and can be parametrized by three real numbers. We make fully
explicit the dynamics of $f$ thanks to two specific functions $\delta$ and
$\phi$ depending on these parameters whose definitions involve Hecke-Mahler
series. As an application, we show that the rotation number of $f$ is rational,
when the three parameters are algebraic numbers.
