Connected McMullen-like Julia sets in a Chebyshev-Halley Family

TL;DR

This work investigates a one-parameter family of rational maps arising from the Chebyshev-Halley root-finding algorithms and analyzes their dynamics near a singular perturbation at $a=0$. By constructing a precise quasiconformal surgery, the authors show that the second iterate $R_a^2$ is dynamically conjugate to a McMullen map $M_\lambda$ on an annulus, yielding a contained copy of $\mathcal{J}(M_\lambda)$ inside $\mathcal{J}(R_a)$ and enabling the realization of all three classical McMullen Julia-set types in a single nearby parameter region $\Lambda$. The Rigidity result for McMullen maps provides a decisive criterion to recognize when a degree-6 map is linearly conjugate to $M_\lambda$, underpinning the surgery. The paper further extends the construction to Chebyshev-Halley methods applied to $z^n-1$, showing a universal mechanism by which singular perturbations near $a=0$ give rise to McMullen-like dynamics and a rich variety of Julia-set phenomena. Overall, the work bridges iterative root-finding dynamics with McMullen’s rational maps through explicit geometric surgery and holomorphic motion techniques, highlighting deep connections between algorithmic dynamics and classical complex dynamics.

Abstract

In this paper we study a one parameter family of rational maps obtained by applying the Chebyshev-Halley root finding algorithms. We show that the dynamics near parameters where the family presents some degeneracy might be understood from the point of view of singular perturbations. More precisely, we relate the dynamics of those maps with the one of the McMullen family $M_λ(z)=z^4 + λ/z^2$, using quasi-conformal surgery.

Figures (11)

• Figure 1: Dynamical planes of $R_a(z)$ for $a=0$ (left) and $a=0.0001$ (right). In both dynamical planes red points represent points converging towards a third root of unity. For $a=0$ (left) black points represent points converging to the super-attracting cycle $\{0,\infty\}$. We mark the third roots of unity $1, e^{2 \pi i/3}, e^{4 \pi i/3}$.
• Figure 2: On the left side we plot the parameter plane of $M_{\lambda}$\ref{['eq:Mcmullen']} and on the right side the parameter plane of $R_a$\ref{['eq:Ra']} near $a=0$.
• Figure 3: In the first column we present the dynamical plane of $M_{\lambda}$\ref{['eq:Mcmullen']} for $\lambda=0.005$ (top) $\lambda=-0.28$ (middle) and $\lambda=-0.455$ (bottom). In the second column the dynamical plane of $R_a$ (see \ref{['eq:Ra']}) for $a=-0.0003$ (top) $a=-0.0164$ (middle) and $a=-0.028$ (bottom) with $\rm{Re}(z)\in (-0.245, 0.245)$ and $\rm{Im}(z)\in (-0.245, 0.245)$. In the dynamical plane of $R_a$ red points represent points converging to a third root of unity while in the McMullen family $M_{\lambda}$ red points represent points converging to infinity.
• Figure 4: Dynamical plane of $R_0$ and sketch of the dynamical rays involved in the construction
• Figure 5: Configuration of the dynamics of the curve $\gamma_0$ and its preimages up to $\gamma_3$. the map $R_0$ maps each curve $\gamma_k$ onto $\gamma_{k-1}$ with degree 2 for $k=1, \ldots, 4$.

Theorems & Definitions (35)

• Theorem
• Corollary
• Definition 1.1
• Theorem : Escape Trichotomy, DLU
• Proposition 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• Definition 3.3
• Lemma 3.4