Double rotations

We consider a map called a double rotation, which is composed of two rotations on a circle. Specifically, a double rotation is a map on the interval [0, 1) that maps x ∈ [0, c) to {x + α}, and x ∈ [c, 1) to {x + β}. Although double rotations are discontinuous and non-invertible in general, we show that almost every double rotation can be reduced to a simple rotation, and the set of parameters such that the double rotation is irreducible to a rotation has a fractal structure. We also examine a characteristic number of double rotations that is called a discharge number. The discharge number as a function of c reflects the fractal structure, and is very complicated.