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Devil's Staircase

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DevilsStaircase

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

theta_(n+1)=theta_n+Omega-K/(2pi)sin(2pitheta_n)
(1)

with K=1. (Since the circle map becomes mode-locked, the map winding number is independent of the initial starting argument theta_0.) At each value of Omega, 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 [0,1] onto [0,1], but is constant almost everywhere (i.e., except on a Cantor set).

For K=1, the measure of quasiperiodic states (Omega irrational) on the Omega-axis has become zero, and the measure of mode-locked state has become 1. The dimension of the Devil's staircase approx 0.8700+/-3.7×10^(-4).

DevilsStaircaseFloor

Another type of devil's staircase occurs for the sum

f(x)=sum_(n=1)^infty(|_nx_|)/(2^n)
(2)

for x in [0,1], where |_x_| 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 x but discontinuous at every rational x. f(x) is irrational iff x is, and if x is irrational, then f(x) is transcendental. If x=p/q is rational, then

f(x)=1/(2^q-1)+sum_(m=1)^infty1/(2^(|_m/x_|)),
(3)

while if x is irrational,

f(x)=sum_(m=1)^infty1/(2^(|_m/x_|)).
(4)

Even more amazingly, for irrational x with simple continued fraction [0,a_1,a_2,...] and convergents p_n/q_n,

f(x)=[0,A_1,A_2,A_3,...],
(5)

where

A_n=2^(q_(n-2))(2^(a_nq_(n-1))-1)/(2^(q_(n-1))-1)
(6)

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

f(phi^(-1))=[0,2^(F_0),2^(F_1),2^(F_2),...],
(7)

where phi is the golden ratio and F_n 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 |_nalpha+gamma_|." 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 |_nalpha+s_| and the Zero of {nalpha+s}." 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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