A transformation of the circle

An investigation of the transformation $T: x\to\lambda\sin x\pmod\pi$ of a circle, which corresponds to a phase transformation encountered in some problems concerning nonlinear vibrations, is carried out. It is proved that if $\lambda=q\pi$ where $q$ is a nonzero integer, there is a measure invariant under the transformation and absolutely continuous with respect to Lebesgue measure.