Some Researches On Complex Dynamic Of The Relaxed Newton’s Method And The Family Of Beardon Maps

For the relaxed Newton’s method of complex exponential function, its special dy-namic properties are summarized and extended. In theory, it is proved that under given conditions, all periodic Fatou components are Jordan domains; and some dynamical properties for the relaxed Newton’s method of special complex exponential functions are depicted. In application, the prescribed set be a superattracting cycle is constructed by the relaxed Newton’s method, and on the basis of that, the same prescribed set be any type cycle is constructed. On the other hand, for every polynomial in some families of polynomials, all its roots or real roots can be found under the iterative of its Newton’s method by choosing a little fixed starting points. The second part of this dissertation focus on the study of connectivity and boundary properties of Fatou components for the family of Beardon maps. It is proved that there exist some Fatou components with different connectivity (all bigger than two). However, the boundaries of these Fatou components consist of Jordan curves under certain conditions.