2006/01/26 by Xavier Buff, Buff, Xavier, Johannes Rueckert +1
Mathematics · #30D05 #37F10 (Primary) #49M15 (Secondary) #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.math/0601644
openalex publication_date 2006/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Newton's root finding method applied to a (transcendental) entire function f:C->C is the iteration of a meromorphic function N. It is well known that if for some starting value z, Newton's method converges to a point x in C, then f has a root at x. We show that in many cases, if an orbit converges to infinity for Newton's method, then f has a `virtual root' at infinity. More precisely, we show that if N has an invariant Baker domain that satisfies some mild assumptions, then 0 is an asymptotic value for f. Conversely, we show that if f has an asymptotic value of logarithmic type at 0, then the singularity over 0 is contained in an invariant Baker domain of N, which we call a virtual immediate basin. We show by way of counterexamples that this is not true for more general types of singularities.