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