Geodesic
Derivatives of circle rotations and similarity of orbits
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 272-284 
A. V. Shutov. Derivatives of circle rotations and similarity of orbits. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 272-284. http://geodesic.mathdoc.fr/item/ZNSL_2004_314_a16/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is proved that the derivative of a circle rotation on an arbitrary interval is either a circle rotation or a noncyclic exchange of three intervals. In the former case, all possible values of the new angle of rotation are computed. It is shown that the restriction of the orbit of a circle rotation to an eigeninterval of differentiation is similar to the orbit of another circle rotation.

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