Connectivity of the Julia set for the Chebyshev-Halley family on degree n polynomials
For mathematicians studying root-finding algorithms, this provides a theoretical criterion for connectivity of Julia sets, but the results are specific to a particular family of polynomials.
The paper provides a criterion guaranteeing simple connectivity of basins of attraction for Chebyshev-Halley root-finding methods, and characterizes parameters for which the Julia set is connected for polynomials z^n+c, showing how increasing n affects dynamics.
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.