Basins of Newton Maps and Asymptotic Values

Newton’s root finding method applied to a (transcendental) entire function f : C → C is the iteration of a meromorphic function Nf . It is well known that if for some starting value z0, Newton’s method converges to a point ξ ∈ C, then f has a root at ξ. We show that in many cases, if an orbit converges to ξ = ∞ for Newton’s method, then f has a ‘virtual root’ at ∞. More precisely, we show that if Nf 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 Nf , which we call a virtual immediate basin. We show by way of counterexamples that this is not true for more general types of singularities.