Newton's method applied to rational functions: Fixed points and Julia sets

TL;DR

This work analyzes Newton maps $N_R$ arising from rational functions $R$ by examining the fixed-point structure, connectivity of Julia sets, and symmetries. Using a blend of conjugacy, scaling, and multiplier analysis, it classifies Newton maps with exactly two attracting fixed points (one exceptional) as conjugate to $N_R$ with $R(z)=\frac{z^d}{p(z)}$ and explores the resulting Julia-set connectivity under multiplier constraints; it also provides explicit classifications for low-degree cases. The paper proves that certain families yield connected Julia sets, including the McMullen maps $f_\lambda(z)=z^m-\frac{\lambda}{z^n}$, whose Newton maps are connected and exhibit rotational symmetry of order $m+n$, while other polynomials like $N_{1/p}$ produce totally disconnected Julia sets in several cases. Overall, all identified connected Julia sets are shown to be locally connected, contributing a detailed structural understanding of Newton dynamics on rational functions and enriching the broader theory of complex dynamics for root-finding maps.

Abstract

For a rational function $R$, let $N_R(z)=z-\frac{R(z)}{R'(z)}.$ Any such $N_R$ is referred to as a Newton map. We determine all the rational functions $R$ for which $N_R$ has exactly two attracting fixed points, one of which is an exceptional point. Further, if all the repelling fixed points of any such Newton map are with multiplier $2$, or the multiplier of the non-exceptional attracting fixed point is at most $\frac{4}{5}$, then its Julia set is shown to be connected. If a polynomial $p$ has exactly two roots, is unicritical but not a monomial, or $p(z)=z(z^n+a)$ for some $a \in \mathbb{C}$ and $n \geq 1$, then we have proved that the Julia set of $N_{\frac{1}{p}}$ is totally disconnected. For the McMullen map $f_λ(z)=z^m - \fracλ{z^n}$, $λ\in \mathbb{C}\setminus \{0\}$ and $m,n \geq 1$, we have proved that the Julia set of $N_{f_λ}$ is connected and is invariant under rotations about the origin of order $m+n$. All the connected Julia sets mentioned above are found to be locally connected.

Figures (4)

• Figure 4: The Julia set of $F_4 (z)=\frac{z(9 i z^3+(3 \sqrt{5}+ 3i)z^2+(-\sqrt{5}+ 2i)z+8\sqrt{5}+ 56i)}{10(\sqrt{5}+ 7i)}$ and $F_2 (z)= \frac{z}{20} (z^4 +z^3+z^2+z+16)$ are shown as the boundary of the yellow and the blue regions in each image.
• Figure 5: The Fatou set of the Newton's method of McMullen maps for $m=1$
• Figure 6: The graph of $N_f (z)=\frac{z((m-1)z^{m+n}+ n+1)}{m z^{m+n}+n}$
• Figure 7: The Fatou set of $N_f (z)=\frac{z((m-1)z^{m+n}+ n+1)}{m z^{m+n}+n}$

Theorems & Definitions (43)

• Theorem 1.1: Shishikura2009
• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Lemma 2.1: Scaling property
• Remark 2.1
• Lemma 2.2
• Remark 2.2