Connectivity of the Julia set for the Chebyshev-Halley family on degree n polynomials

TL;DR

This paper investigates the connectivity of Julia sets for the Chebyshev-Halley family applied to degree n polynomials, providing criteria for simple connectivity of basins of attraction and analyzing the impact of increasing n.

Contribution

It introduces a new criterion ensuring simple connectivity of basins for Chebyshev-Halley methods on degree n polynomials, characterizing parameters for connected Julia sets.

Findings

Criteria for simple connectivity of basins of attraction.

Parameter characterization for connected Julia sets.

Effect of increasing degree n on dynamics.

Abstract

We study the Chebyshev-Halley family of root finding algorithms from the point of view of holomorphic dynamics. Numerical experiments show that the speed of convergence to the roots may be slower when the basins of attraction are not simply connected. In this paper we provide a criterion which guarantees the simple connectivity of the basins of attraction of the roots. We use the criterion for the Chebyshev-Halley methods applied to the degree $n$ polynomials $z^n+c$, obtaining a characterization of the parameters for which all Fatou components are simply connected and, therefore, the Julia set is connected. We also study how increasing $n$ affects the dynamics.