On the dynamics of Halley's method

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.