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On the quadratic mapping \(z\rightarrow z^{2}-\mu \) for complex \(\mu \) and \(z\): the fractal structure of its set, and scaling. (English) Zbl 1194.30028
Summary: For each complex \(\mu \), denote by \(\mathcal F(\mu )\) the largest bounded set in the complex plane that is invariant under the action of the mapping \(z\rightarrow z^{2}-\mu \). The author [in: Nonlinear dynamics, int. Conf., New York 1979, Ann. N.Y. Acad. Sci. 357, 249–259 (1980; Zbl 0478.58017); The fractal geometry of nature (1982; Zbl 0504.28001) (Chap. 19)] reported various remarkable properties of the \(\mathcal M\) set (the set of those values of the complex \(\mu \) for which \(\mathcal F(\mu )\) contains domains) and of the closure \(\mathcal M^{*}\) of \(\mathcal M\) . The goals of the present work are as follows. A) To restate some previously reported properties of \(\mathcal F(\mu )\), \(\mathcal M\) and \(\mathcal M^{*}\) in new ways, and to report new observations. B) To deduce some known properties of the mapping \(f\) for real \(\mu \) and \(z\), with \(\mu\in ]-1/4, 2[\) and \(z\in ]-1/2, -1/2\sqrt{1+4\mu}, 1/2+1/2\sqrt{1+4\mu}[\). In many ways, the properties of the transformation \(f\) are easier to grasp in the complex plane than in an interval. (This exemplifies the saying that “when one wishes to simplify a theory, one should complexify the variables”,) C) To serve as introduction to some recent pure mathematical work triggered by Mandelbrot 1980. Further pure mathematical work is strongly urged.

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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