A plot of the map winding number resulting from mode locking as a function of for the circle map

with . (Since the circle map becomes mode-locked, the map winding number is independent of the initial starting argument .) At each value of , the map winding number is some rational number. The result is a monotonic increasing "staircase" for which the simplest rational numbers have the largest steps. The Devil's staircase continuously maps the interval onto , but is constant almost everywhere (i.e., except on a Cantor set).

For , the measure of quasiperiodic states ( irrational) on the -axis has become zero, and the measure of mode-locked state has become 1. The dimension of the Devil's staircase .

Another type of devil's staircase occurs for the sum

for , where is the floor function (Böhmer 1926ab; Kuipers and Niederreiter 1974, p. 10; Danilov 1974; Adams 1977; Davison 1977; Bowman 1988; Borwein and Borwein 1993; Bowman 1995; Bailey and Crandall 2001; Bailey and Crandall 2003). This function is monotone increasing and continuous at every irrational but discontinuous at every rational . is irrational iff is, and if is irrational, then is transcendental. If is rational, then

while if is irrational,

Even more amazingly, for irrational with simple continued fraction and convergents ,

where

(Bailey and Crandall 2001). This gives the beautiful relation to the Rabbit constant

where is the golden ratio and is a Fibonacci number.