FRACTALS from NONLINEAR IFSs of the COMPLEX MAPPING FAMILY ƒ(z) = zⁿ + c

To generate exotic fractals, we investigate the construction of nonlinear iterated function system (IFS) using the complex mapping family f(z)= zⁿ+c (|n| ≥ 2, 3,...). 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 Dn-1 symmetry for n ≥ 2 and with Z_|n|+1 or D_|n|+1 symmetry for n ≤-2.