| The technique of matings of polynomials was developed by Douady and Hubbard, and coupled with work of Thurston, it has been widely used to study post-critically finite rational maps. Tan Lei studied the family of cubic Newton's maps using this technique. In this thesis, we study the family of Newton's method applied to the quartic polynomials of the form ;In this parameter space, we observe the presence of tricorn-like figures, which suggests that the free critical points of the rational maps exhibit bitransitive behavior. We divide the parameter space into two cases depending on the mating possibilities. We call the first case 3 + 1 matings. Here, a rational map is Thurston equivalent to a mating of two polynomials where one polynomial has three superattracting fixed points and the other has one. We prove that a post-critically finite rational map from this Newton's method family is Thurston equivalent to a mating of polynomials if and only if the two free critical points of the rational map do not eventually map onto the two complex conjugate roots. The tricorn-like figures belong in this category. We call the second case 2 + 2 matings, indicating that each of the two mating polynomials has two superattracting fixed points. We show that there is one Newton's rational map in the parameter space which is a 2 + 2 mating. Also, we show examples to suggest strongly that there are no other Newton's maps in this family that are 2 + 2 matings. |
