On the dynamics of Halley's method

TL;DR

This paper investigates the complex dynamics of Halley's method for polynomial root-finding, analyzing the structure of its Julia and Fatou sets, and establishing properties like connectivity, symmetry, and convergence for various polynomial classes.

Contribution

It provides new insights into the global dynamics of Halley's method, including proofs of connected Julia sets and symmetry properties for specific polynomial classes.

Findings

Julia set of Halley's method is connected for certain polynomials

Immediate basins are unbounded and share symmetry with the polynomial

Results extend to broader classes of polynomials, including bounded basins

Abstract

In this article, we study the global dynamics of Halley's method applied to complex polynomials. Specifically, we analyze the structure and connectivity of the Julia set of this method. The convergence behavior, symmetry properties, and topological features of the corresponding Fatou and Julia sets are studied for various classes of polynomials, including unicritical, cubic, and quartic polynomials with non-trivial symmetry groups. In particular, we prove that the Halley's method $H_p$ is convergent, its Julia set is connected, the immediate basins are unbounded and the symmetry group of it coincides with that of the polynomial whenever $p$ belongs to one of the above classes. We further extend our results to a broader class of polynomials. It is shown that the immediate basin of the Halley's method $H_p$ corresponding to a root of $p$ can be bounded. We also make some remarks on the dynamics of the Halley's method applied to a cubic polynomial in general.