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
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"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. USSR12,
524-527, 1974.Davison, J. L. "A Series and Its Associated
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B. B. The
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