Geodesic
A transformation of the circle
Matematičeskie zametki, Tome 8 (1970) no. 2, pp. 205-216 
L. A. Bunimovich. A transformation of the circle. Matematičeskie zametki, Tome 8 (1970) no. 2, pp. 205-216. http://geodesic.mathdoc.fr/item/MZM_1970_8_2_a8/
@article{MZM_1970_8_2_a8,
     author = {L. A. Bunimovich},
     title = {A transformation of the circle},
     journal = {Matemati\v{c}eskie zametki},
     pages = {205--216},
     year = {1970},
     volume = {8},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_2_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.