Two-colour rotations of the unit circle

We consider two-colour, or double, rotations $S_{(\alpha,\beta,\varepsilon)}(x)$ of the unit circle $C$ the colouring of which depends on a continuous parameter $\varepsilon\in C$ and each area of which is given its own rotation angle, $\alpha$ or $\beta$. We choose as a model the one-parameter family of two-colour rotations $S_\varepsilon(x)=S_{(2\tau,\tau,\varepsilon)}(x)$, where $\tau=(1+\sqrt{5}\,)/2$ is the golden ratio, which have rotation rank $d=2$. It is proved that the first-return map $S_\varepsilon|\mathrm{Att}_\varepsilon$ (the restriction of the rotation $S_\varepsilon(x)$ to its attractor $\mathrm{Att}_\varepsilon$) is isomorphic to the integral map $T_\varepsilon=T(S^{\pm1},d_\varepsilon)$ constructed from the simple rotation $S$ of the circle through the angle $\pm \tau$ and some piecewise-constant function $d_\varepsilon$. An exact formula is obtained for the function $\nu(\varepsilon)$ of frequency distribution of points of the orbits under the action of $S_\varepsilon$.