Dynamics Of 2-Interval Piecewise Affine Maps And Hecke-Mahler Series

Let f : [0,1) -> [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, whenever the three parameters are all algebraic numbers, extending thus the main result of [16] dealing with the particular case of 2-interval piecewise affine contractions with constant slope.