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A general view of pseudoharmonics and pseudoantiharmonics to calculate external arguments of Douady and Hubbard. (English) Zbl 1176.30072

Summary: Harmonics give us a compact formula and a powerful tool in order to calculate the external arguments of the last appearance hyperbolic components and Misiurewicz points of the Mandelbrot set in some particular cases. Antiharmonics seem however to have no application. In this paper, we give a general view of pseudoharmonics and pseudoantiharmonics, as a generalization of harmonics and antiharmonics. Pseudoharmonics turn out to be a more powerful tool than harmonics since they allow the calculation of external arguments of the Mandelbrot set in many more cases. Likewise, unlike antiharmonics, pseudoantiharmonics turn out to be a powerful tool to calculate external arguments of the Mandelbrot set in some cases.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
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