Geodesic
Renormalization on the one-dimansion torus
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 142-154

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Renormalization formulas connecting the points of the orbits $O^\pm_0$, i.e., the sequences generated by an irrational shift on the half-open interval $[0,1)$ or $(0,1]$, and the points of the orbits' derivatives $d^mO^\pm_0$, i.e., the sequences obtained by restricting $O^\pm_0$ to half-open intervals of smaller length, are obtained. 
@article{ZNSL_2004_314_a8,
     author = {N. N. Manuylov},
     title = {Renormalization on the one-dimansion torus},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {142--154},
     publisher = {mathdoc},
     volume = {314},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_314_a8/}
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N. N. Manuylov. Renormalization on the one-dimansion torus. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 142-154. http://geodesic.mathdoc.fr/item/ZNSL_2004_314_a8/