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Fractals from nonlinear IFSs of the complex mapping family \(f(z) = z^n + c\). (English) Zbl 1433.28010

Summary: To generate exotic fractals, we investigate the construction of nonlinear iterated function system (IFS) using the complex mapping family \(f(z) = z^n + c (| n | \geq 2, 3, \ldots )\). A set of \(c\)-values is chosen from the period-1 bulb of the Mandelbrot set, so that each mapping has an attracting fixed point in the dynamic plane. Computer experiments show that a set of arbitrarily chosen \(c\)-values may not be able to generate a fractal. We prove a sufficient condition that if the \(c\)-values are chosen from a specific region related to a circle in the period-1 bulb, the nonlinear IFS with such complex mappings is able to generate exotic fractal. Furthermore, if the set of \(c\)-values possesses a specific symmetry in the Mandelbrot set, then the fractal also exhibits the same symmetry. We present a method of generating aesthetic fractals with \(Z_{n - 1}\) or \(D_{n - 1}\) symmetry for \(n \geq 2\) and with \(Z_{| n | + 1}\) or \(D_{| n | + 1}\) symmetry for \(n \leq - 2\).

MSC:

28A80 Fractals
37F44 Holomorphic families of dynamical systems; holomorphic motions; semigroups of holomorphic maps
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