Abstract
AbstractWe study the Chebyshev–Halley methods applied to the family of polynomials$$f_{n,c}(z)=z^n+c$$fn,c(z)=zn+c, for$$n\ge 2$$n≥2and$$c\in \mathbb {C}^{*}$$c∈C∗. We prove the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. We also prove that, for$$n \ge 2$$n≥2, the corresponding dynamical plane contains a connected component of the Julia set, which is a quasiconformal deformation of the Julia set of the map obtained by applying Newton’s method to$$f_{n,-1}$$fn,-1.
Publisher
Springer Science and Business Media LLC
Reference18 articles.
1. Advances in iterative methods for nonlinear equations, SEMA SIMAI Springer Series, vol. 10. Springer, Cham (ISBN 978-3-319-39227-1/hbk; 978-3-319-39228-8/ebook), pp. v + 286 (2016)
2. Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. (N.S.) 10, 3–35 (2004) (English)
3. Barański, K., Fagella, N., Jarque, X., Karpińska, B.: On the connectivity of the Julia sets of meromorphic functions. Invent. Math. 198(3), 591–636 (2014) (English)
4. Beardon, A.F.: Iteration of Rational Functions, Graduate Texts in Mathematics, vol. 132. Springer, New York (1991). Complex analytic dynamical systems
5. Branner, B., Fagella, N.: Quasiconformal surgery in holomorphic dynamics, Cambridge Studies in Advanced Mathematics, vol. 141. Cambridge University Press, Cambridge (2014)