Misc,
A formula for the fractal dimension d approx. 0.87 of the Cantorian set underlying the Devil's staircase associated with the Circle Map
(2007)cite arxiv:0711.2706 Comment: 53 pages double spaced, 1 figure.
Abstract
The Cantor set complementary to the Devil's Staircase associated with the
Circle Map has a fractal dimension d approximately equal to 0.87, a value that
is universal for a wide range of maps, such results being of a numerical
character. In this paper we deduce a formula for such dimensional value. The
Devil's Staircase associated with the Circle Map is a function that transforms
horizontal unit interval I onto vertical I, and is endowed with the
Farey-Brocot (F-B) structure in the vertical axis via the rational heights of
stability intervals. The underlying Cantor-dust fractal set Omega in the
horizontal axis --Omega contained in I, with fractal dimension d(Omega) approx.
0.87-- has a natural covering with segments that also follow the F-B hierarchy:
therefore, the staircase associates vertical I (of unit dimension) with
horizontal Omega in I (of dimension approx. 0.87), i.e. it selects a certain
subset Omega of I, both sets F- B structured, the selected Omega with smaller
dimension than that of I. Hence, the structure of the staircase mirrors the F-
B hierarchy. In this paper we consider the subset Omega-F-B of I that
concentrates the measure induced by the F-B partition and calculate its
Hausdorff dimension, i.e. the entropic or information dimension of the F-B
measure, and show that it coincides with d(Omega) approx. 0.87. Hence, this
dimensional value stems from the F-B structure, and we draw conclusions and
conjectures from this fact. Finally, we calculate the statistical "Euclidean"
dimension (based on the ordinary Lebesgue measure) of the F-B partition, and we
show that it is the same as d(Omega-F-B), which permits conjecturing on the
universality of the dimensional value d approximately equal to 0.87.
