Global dynamics of Newton’s method for complex polynomials

Abstract

[en] Newton’s method, as a root-finding algorithm, has been used since ancient times to solve daily problems. Nevertheless, it was not until the second half of the nineteenth century that it began being studied as a dynamical system in the complex plane. Following this path, the main goal of this thesis is to understand and prove, using recently developed techniques, Shishikura’s result on the connectivity of the Julia set of the Newton map of polynomials. To do so, we first present a set of preliminary tools that contain normal families, conformal representations and proper maps, among others. It is followed by a study of rational complex dynamical systems, some results on the existence of fixed points of meromorphic maps and it is concluded by what is the cornerstone of this project: the proof of the connectivity of the Julia set of Newton maps of polynomials.