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Devil's staircase is 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.

See also

Cantor Function, Circle Map, Map Winding Number, Minkowski's Question Mark Function, Rabbit Constant

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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.

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Devil's Staircase

Cite this as:

Weisstein, Eric W. "Devil's Staircase." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DevilsStaircase.html

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