We construct an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative.On the other hand, we show that for many functions in B, in particular those of finite order, every escaping point can be connected to ∞ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

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We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps.A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to ∞ through a finite chain of such components.

We construct an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative.On the other hand, we show that for many functions in B, in particular those of finite order, every escaping point can be connected to ∞ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

Abstract. Newton's root finding method applied to a (transcendental) entire function f : C → C is the iteration of a meromorphic function N f . 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 N f 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 f , which we call a virtual immediate basin. We show by way of counterexamples that this is not true for more general types of singularities.