×

Local topology of cubic Newton methods: Dynamical plan. (Topologie locale des méthodes de Newton cubiques: Plan dynamique.) (French. Abridged English version) Zbl 0924.58084

The paper under review is concerned with the local connectivity of the boundary of the immediate basins, and of the Julia set of a cubic Newton map. Let \(N\) be a cubic Newton map, conformally conjugate to Newton’s method \(N_P\) of \(P\) a cubic polynomial with simple roots. The map \(N\) has four critical points, three of which are roots of \(P\) and the fourth of which is a free critical point with various dynamics under iterations of \(N\). The author obtains the following results:
Theorem 1. The boundary of the immediate basin of each root is locally connected.
Theorem 2. The Julia set of \(N\) is locally connected, provided that either
\(\bullet\) \(N\) has no irrational indifferent periodic point, or
\(\bullet\) \(N\) has no Siegel disc and the forward orbit of free critical points does not accumulate on the boundary of the immediate basins.
Corollary. There exists a cubic Newton map with a Cremer point and locally connected Julia set.
Work of L. Tan and Y.-C. Lee referred to in this paper, takes a part in Theorem 2.

MSC:

37F99 Dynamical systems over complex numbers
PDFBibTeX XMLCite
Full Text: DOI