Devil's Staircase

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.

REFERENCES:

Adams, W. W. "A Remarkable Class of Continued Fractions." Proc. Amer. Math. Soc. 65, 194-198, 1977.

Bailey, D. H. and Crandall, R. E. "On the Random Character of Fundamental Constant Expansions." Exper. Math. 10, 175-190, 2001.

Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.

Böhmer, P. E. "Über die Transcendenz gewisser dyadischer Brüche." Math. Ann. 96, 367-377, 1926a.

Böhmer, P. E. Erratum to "Über die Transcendenz gewisser dyadischer Brüche." Math. Ann. 96, 735, 1926b.

Borwein, J. and Borwein, P. "On the Generating Function of the Integer Part of ." J. Number Th. 43, 293-318, 1993.

Bowman, D. "A New Generalization of Davison's Theorem." Fib. Quart. 26, 40-45, 1988.

Bowman, D. "Approximation of  and the Zero of ." J. Number Th. 50, 128-144, 1995.

Danilov, L. V. "Some Classes of Transcendental Numbers." Math. Notes Acad. Sci. USSR 12, 524-527, 1974.

Davison, J. L. "A Series and Its Associated Continued Fraction." Proc. Amer. Math. Soc. 63, 29-32, 1977.

Devaney, R. L. An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley, pp. 109-110, 1987.

Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. New York: Wiley, 1974.

Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, pp. 82-83 and 286-287, 1983.

Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993.

Rasband, S. N. "The Circle Map and the Devil's Staircase." §6.5 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 128-132, 1990.