Rotation number of 2-interval piecewise affine maps

We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps f_p are parametrized by a quintuple p of real numbers satisfying inequations. Viewing f_p as a circle map, we show that it has a rotation number ρ (f_p) and we compute ρ (f_p) as a function of p in terms of Hecke–Mahler series. As a corollary, we prove that ρ (f_p) is a rational number when the components of p are algebraic numbers.