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 is an element of [0, c) to {x + alpha}, and x is an element of [c, 1) to {x + beta}. Although double rotations are discontinuous and noninvertible in general, we show that almost every double rotation can be reduced to a simple rotation, and the set of the parameter values such that the double rotation is irreducible to a rotation has a fractal structure. We also examine a characteristic number of a double rotation, which is called a discharge number. The graph of the discharge number as a function of c reflects the fractal structure, and is very complicated.