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

Abstract

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.