The “normalized radical” of the ℳ-set

Benoit B. Mandelbrot
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Benoit B. Mandelbrot: Yale University, Mathematics Department

Chapter C7 in Fractals and Chaos, 2004, pp 100-109 from Springer

Abstract: 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 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaaeWaWdaeaapeGaeuOPdyKaaiikaiaad6gacaGGPaGaamOBa8aa % daahaaWcbeqaa8qacqGHsislcaaIYaGaamiraaaakiabg2da9iaaig % daaSWdaeaapeGaaGOmaaWdaeaapeGaeyOhIukaniabggHiLdaaaa!43E2! $$ \sum\nolimits_2^\infty {\Phi (n){n^{ - 2D}} = 1} $$ , where Φ(n) is Euler’s number-theoretic function. A rough first approximation is the solution D = 1.239375 of the equation % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaaeWaWdaeaapeGaamOBa8aadaahaaWcbeqaaiaaigdacqGHsisl % caaIYaGaamiraaaak8qacqGH9aqpcqaH2oGEcaGGOaGaaGOmaiaads % eacqGHsislcaaIXaGaaiykaiabgkHiTiaaigdacqGH9aqpcqaHapaC % daahaaWcbeqaaiaaikdaaaGccaGGVaGaaGOnaaWcpaqaa8qacaaIYa % aapaqaa8qacqGHEisPa0GaeyyeIuoaaaa!4D30! $$ \sum\nolimits_2^\infty {{n^{1 - 2D}} = \zeta (2D - 1) - 1 = {\pi ^2}/6} $$ , 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.

Keywords: Fractal Dimension; Generation Atom; Hausdorff Measure; Riemann Zeta Function; Closed Disc (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-1-4757-4017-2_7

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