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 f p are parametrized by a quintuple p p of real numbers satisfying inequations. Viewing f_p f p as a circle map, we show that it has a rotation number ρ (f_p) ρ ( f p ) and we compute ρ (f_p) ρ ( f p ) as a function of p p in terms of Hecke–Mahler series. As a corollary, we prove that ρ (f_p) ρ ( f p ) is a rational number when the components of p p are algebraic numbers.