Two-colour rotations of the unit circle

Abstract

We consider two-colour, or double, rotations of the unit circle the colouring of which depends on a continuous parameter and each area of which is given its own rotation angle, or . We choose as a model the one-parameter family of two-colour rotations , where is the golden ratio, which have rotation rank . It is proved that the first-return map (the restriction of the rotation to its attractor ) is isomorphic to the integral map constructed from the simple rotation of the circle through the angle and some piecewise-constant function . An exact formula is obtained for the function of frequency distribution of points of the orbits under the action of .