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Generalized Mandelbrot sets arise in perturbed (non-analytic) 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 non-divergent 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.

Smooth Decomposition of Generalized Fatou Set Explains Smooth Structure in Generalized Mandelbrot Set