M and J sets from Newton's transformation of the transcendental mapping F(z) = e(z(w)+c) with vcps

Newton's transformation f_w(z) = z - 1/(wz^w-1) containing only one complex parameter w (w ≠ 0 or 1) is constructed from the transcendental mapping F(z) = e(^z(w)+c). Although the number of critical points of f_w(z) is countably infinite, a method based on the Valid Critical Point Set, vcps = {z_k ∈ C| - π <arg(z_k) ≤ π, f_w′(z_k) = 0, k ∈ Z}, is discussed for the generation of the generalized Mandelbrot set M of f_w(z). The petal fragments, the multi-period fragments, and the classical "Mandelbrot set" fragments are found in M. The dynamical characteristics of f_w(z) for different values of w are analyzed. The relationship between the parameter w in a classical "Mandelbrot set" fragment and the structure of the corresponding filled-in Julia set is investigated. A large number of novel images of filled-in Julia sets are generated based on the rate of convergence of orbits. © 2002 Published by Elsevier Science Ltd.