The “normalized radical” of the ℳ-set

Abstract

A “normalized radical” ℛ of the ℳ-set is defined as the shape that satisfies exactly all the self-similarity properties that hold approximately for the molecules of the ℳ-set of the quadratic map. Explicit constructions show that the complement of ℛ is a σ-lune, and prove that the ℛ-set does not self-overlap. The fractal dimension D of the boundary of ℛ is shown to satisfy \( \sum\nolimits_2^\infty {\Phi (n){n^{ - 2D}} = 1} \), where Φ(n) is Euler’s number-theoretic function.

A rough first approximation is the solution D = 1.239375 of the equation \( \sum\nolimits_2^\infty {{n^{1 - 2D}} = \zeta (2D - 1) - 1 = {\pi ^2}/6} \), where ζ is the Riemann zeta function. A less elegant but doubtless closer second approximation is D=1.234802. The same D applies to the ℳ-sets of other maps in the same class of universality.

Interesting “rank-size” probability distributions are introduced.