Renormalization on the one-dimansion torus

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.