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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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##### References:
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