Devil's staircase in a one-dimensional mapping (Q1080242)

We give a one-dimensional mapping which is a simple example that the periodic orbits show an arithmetic furcation as a function of a parameter characterizing the mapping. The mapping is a piecewise linear function which consists of three parts, that is, a line with slope 1, a line with slope 0 and a line with slope \(a>1\). When the frequency is defined by the ratio of the number of times of visiting the lines with slope a and with slope 0 within a period to the period, the frequency takes on the elements of Farey's set and behaves as a complete devil's staircase as a function of a parameter.