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Geometry of the antennas in the Mandelbrot set. (English) Zbl 1072.37506

Summary: In the Mandelbrot set, the bulbs attached directly to the main cardioid are called the p/q-bulbs. The reason for this is that the largest component of the interior of these bulbs consists of c-values for which the quadratic function \(Q_c(z) = z^2 + c\) admits an attracting cycle with rotation number p/q. We give a geometric method to read off p/q from the geometry of the antenna attached to the bulb.

MSC:

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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References:

[1] DOI: 10.2307/2589552 · Zbl 1076.37512
[2] LaVaurs P., C. R. Acad. Sci. Paris Sér. I Math. 303 pp 143– (1986)
[3] Atela P., Ergod. Th. Dynam. Syst. 12 pp 401– (1991)
[4] Douady A., Notes Rep. Math. Sci. Eng. 2 pp 155– (1986)
[5] Farey J., Phil. Mag. J. London 47 pp 385– (1816)
[6] DOI: 10.1142/S0218348X95000564 · Zbl 0929.37016
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