Analysis Seminar

We will give a brief summary of the problem of the connectivity of the Julia set for certain rational or meromorphic maps related to the number of weakly repelling fixed points. From Shishikura's quasi-conformal approach for rational iteration to the meromorphic scenario. In the second part of the talk we will present the key lines of a unified proof of the following result: Let $g$ is a holomorphic function of the complex plane of degree larger than 1 (polynomial or transcendental) and let $N_g$ be its Newton's method. Then the Julia set of $N_g$ is a connected subset of the Rimann sphere, or equivalently, every Fatou component of $N_g$ is a simply connected subset of the plane.