Connectivity of Julia sets of Newton maps: A unified approach

TL;DR

This paper provides a unified, direct proof that the Julia set of Newton's method applied to any holomorphic function in the complex plane is connected, consolidating previous case-by-case proofs into a single approach.

Contribution

It introduces a unified, self-contained proof establishing the connectivity of Julia sets for all Newton maps of holomorphic functions, simplifying and generalizing prior results.

Findings

Julia sets of Newton maps are connected for all holomorphic functions

Unified proof covers polynomial and transcendental cases

Simplifies understanding of Newton map dynamics

Abstract

In this paper we give a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function of the complex plane (a polynomial of degree large than $1$ or an entire transcendental function) is connected. The result was recently completed by the authors' previous work, as a consequence of a more general theorem whose proof spreads among many papers, which consider separately a number of particular cases for rational and transcendental maps, and use a variety of techniques. In this note we present a unified, direct and reasonably self-contained proof which works for all situations alike.