@article{8dd38108f7b74679a16f5a283132b1fa, title = "M and J sets from Newton's transformation of the transcendental mapping F(z) = e(z(w)+c) with vcps", abstract = "Newton's transformation fw(z) = z - 1/(wzw-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 fw(z) is countably infinite, a method based on the Valid Critical Point Set, vcps = {zk ∈ C| - π k) ≤ π, fw′(zk) = 0, k ∈ Z}, is discussed for the generation of the generalized Mandelbrot set M of fw(z). The petal fragments, the multi-period fragments, and the classical {"}Mandelbrot set{"} fragments are found in M. The dynamical characteristics of fw(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. {\textcopyright} 2002 Published by Elsevier Science Ltd.", keywords = "Chaos, Fractal, Julia set, Mandelbrot set, Newton's transformation", author = "Ning Chen and Zhu, {X. L.} and Chung, {K. W.}", year = "2002", month = apr, doi = "10.1016/S0097-8493(01)00185-6", language = "English", volume = "26", pages = "371--383", journal = "Computers and Graphics (Pergamon)", issn = "0097-8493", publisher = "Pergamon Press", number = "2", }