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

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.

The Julia sets of Chebyshev's method with small degrees

“…2. If p is not generic then its Chebyshev's method is conjugate to 5z 4 +15z 3 +24z 2 +22z+6 9(z+1) 3 and its Julia set is connected.…”

“…3 on D r (by Theorem 9.3,[8]), and D r 3 ⊂ D r . It also follows that C λ : U → V is a proper map of degree three.…”

“…In fact, C p = H 0 p . The dynamics of the Chebyshev-Halley family applied to unicritical polynomials z → z n − 1, n ∈ N is investigated in [3]. A necessary and sufficient condition for disconnected Julia sets is found.…”

Newton-Like Components in the Chebyshev–Halley Family of Degree n Polynomials

“…Previously, Campos, Canela, and Vindel have studied the Chebyshev-Halley family applied to f n,c (z) = z n + c, c ∈ C * (see [6,7]). The maps obtained by applying the Chebyshev-Halley family to f n,c are all conjugated to the map obtained by applying the Chebyshev-Halley family to Lemma 2.1).…”

Dynamics of a Family of Rational Operators of Arbitrary Degree

“…The dynamical properties related to an iterative method give important information about its stability. In recent studies, many authors (see [1,7,8,10,14], for example) have found interesting results from a dynamical point of view. One of the main interests in these papers has been the study of the parameter spaces associated to the families of iterative methods applied on low degree polynomials, which allows to distinguish the different dynamical behaviour.…”