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

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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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     author = {A. V. Shutov},
     title = {Derivatives of circle rotations and similarity of orbits},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {272--284},
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     volume = {314},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_314_a16/}
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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/