The Connectivity of the Julia Set and Fixed Points Mitsuhiro Shishikura

For a non-constant complex polynomial P (z), the rational map NP : Ĉ → Ĉ defined as

NP : z → z − P (z) P ′(z)

is called Newton’s method for P . The iteration of NP gives an algorithm for finding the roots of P . All fixed points except one (namely ∞) are roots of P and are attracting for NP . One of the purposes of this paper is to prove that the Julia set of NP is connected.