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arXiv · 0711.2706

A formula for the fractal dimension d approx. 0.87 of the Cantorian set underlying the Devil's staircase associated with the Circle Map

M. N. Piacquadio Losada
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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.

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M. N. Piacquadio Losada. 2007-11-17. A formula for the fractal dimension d approx. 0.87 of the Cantorian set underlying the Devil's staircase associated with the Circle Map. https://arxiv.org/abs/0711.2706

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