The boundless area of perturbed map generated by using relaxed Newton's method

Although the (relaxed) Newton's method is widely used not only in mathematics but also in applied sciences, only few attempts have been made to provide a better and global understanding of the dynamics of (relaxed) Newton's method. Therefore, it is necessary to combine the classical local studies of the (relaxed) Newton's method with the modern global theory of iterative process in complex dynamics. In this paper, we investigate the behavior of a certain very special and beautiful class of dynamical systems known as entire transcendental functions. We pay special attention to the basin of attraction of roots that have boundless area for certain rational maps. Those are constructed via relaxed Newton's method, perturbing the Multibrot set polynomial by an exponential non-constant complex polynomial with degree less than 2.