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

(1)

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

(2)

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

(3)

while if is irrational,

(4)

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

(5)

where

(6)

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

(7)

where is the golden ratio and is a Fibonacci number.