Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series

Let \begin{document}$ f : [0,1)\rightarrow [0,1) $\end{document} be a id="M2">\begin{document}$ 2 $\end{document}-interval piecewise affine increasing map which is injective but not surjective. Such id="M3">\begin{document}$ has rotation number and can parametrized by three real numbers. We make fully explicit the dynamics of id="M4">\begin{document}$ thanks to two specific functions id="M5">\begin{document}$ {\boldsymbol{\delta}} id="M6">\begin{document}$ \phi depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that id="M7">\begin{document}$ rational, whenever are all algebraic numbers, extending thus main result [16] dealing with particular case id="M8">\begin{document}$ contractions constant slope.