<f>M</f> and <f>J</f> sets from Newton's transformation of the transcendental mapping <f>F(z)=ezw+c</f> with vcps

Newton''s transformation <f>fw(z)=z−1/(wzw−1)</f> containing only one complex parameter <f>w</f> (<f>w≠0</f> or 1) is constructed from the transcendental mapping <f>F(z)=ezw+c</f>. Although the number of critical points of <f>fw(z)</f> is countably infinite, a method based on the Valid Critical Point Set, <f>vcps={<fen>zk∈C<cp type="vb" style="s"></fen>−π<arg(zk)&les;π, f′w(zk)=0, k∈Z}</f>, is discussed for the generation of the generalized Mandelbrot set <f>M</f> of <f>fw(z)</f>. The petal fragments, the multi-period fragments, and the classical “Mandelbrot set” fragments are found in <f>M</f>. The dynamical characteristics of <f>fw(z)</f> for different values of <f>w</f> are analyzed. The relationship between the parameter <f>w</f> 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. [Copyright &y& Elsevier]