Bad Cycles, Blowup Points, and Baby Mandelbrot Sets

This dissertation focuses on two problems in complex dynamics. The first relates to Newton\u27s method, while the second is a study of the particular dynamical properties of a one-parameter family of rational maps. Newton\u27s method is one of the most commonly used algorithms to find the roots of a polynomial. For a fixed polynomial the roots of the polynomial are the attracting fixed points for the Newton map. It follows that for each root there is an open set of points in the complex plane which will converge to the root under iteration of the Newton map. Unfortunately, for some polynomials, the Newton map may possess other (extraneous) attracting cycles, and these also possess an open set of points that converge to the cycle. The first question considered is How bad can this situation be? It is shown that any pre-specified cycle of length n is an extraneous cycle for the Newton map of a polynomial of degree at most n + 1. This estimate is sharp, and the best previous estimate for the degree was 2n. Alternatively if only the cycle length, n ≥ 2, and the degree of the polynomial, d ≥ 3, are chosen then a polynomial is constructed for which the Newton map possesses an extraneous cycle of length n. Similar results are obtained for the relaxed Newton\u27s method, a related root-finding algorithm. The second focus is to study the general dynamical properties of a family of rational maps similar to a family studied by Devaney, Holzer and Uminsky. A conjugacy between this family and a family of relaxed Newton maps is produced. This overcomes some of the difficulties in analyzing the dynamics, as results about either family, through the conjugacy, immediately obtain the corresponding results for the other. It is shown that the parameter planes for these families possess blowup points, homeomorphic copies of the Mandelbrot set and other interesting features