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Smooth decomposition of generalized Fatou set explains smooth structure in generalized Mandelbrot set. (English) Zbl 0987.37500

Summary: Generalized Mandelbrot sets arise in perturbed (nonanalytic) versions of the complex logistic map. Numerically, it contains smooth portions as shown previously. To exclude that this result is specific to particular initial conditions only, the structure of the analogue to the Fatou set is looked at in the region in question. The set of nondivergent points is being ‘eaten up’ by a smooth invading boundary. Therefore, the same type of decomposition applies, independent of position in parameter space, in the region in question.

MSC:

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