Virtual Immediate Basins of Newton Maps and Asymptotic Values

Abstract

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.