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

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 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.