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

TL;DR
This paper derives a formula for the universal fractal dimension (~0.87) of the Cantor set associated with the Devil's Staircase in the Circle Map, linking it to Farey-Brocot structure and measure theory.
Contribution
It provides a mathematical formula for the fractal dimension of the Cantor set underlying the Devil's Staircase, connecting it to Farey-Brocot hierarchy and measure-based dimensions.
Findings
The fractal dimension d is approximately 0.87.
The dimension coincides with the Hausdorff and information dimensions of the Farey-Brocot measure.
The dimension value appears to be universal across related maps.
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…
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Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Mathematical Theories and Applications · Fractal and DNA sequence analysis
