Johannes Rückert

• We study transcendental and rational mappings that arise as Newton maps of entire functions. Our first result is that "in between" any two accesses to infinity of an immediate basin, a Newton map exhibits either another immediate basin, a virtual immediate basin or infinitely many preimages of some point. This result is joint work with Dierk Schleicher and allows to locate virtual immediate basins. An important corollary is a proof of the folklore result that for Newton maps of polynomials, every complementary component of an immediate basin contains another immediate basin. Our second main result, which is joint work with Xavier Buff, shows an interesting connection between virtual immediate basins of the Newton map N_f and asymptotic values of f, answering a 2003 question of Douady: in many cases, 0 is an asymptotic value of f if N_f has a virtual immediate basin. Conversely, if f has an asymptotic value of logarithmic type at 0, then N_f has a virtual immediate basin. We show by way of counterexamples that this is not true for other types of asymptotic values. Our third main result gives a combinatorial classification of a class of Newton maps of polynomials. Let N be the Newton map of a polynomial such that all critical points of N land on a fixed point after finitely many iterations. In this case, we construct a graph that characterizes N uniquely up to Möbius conjugation. Conversely, we show that every graph with an associated map that satisfies several natural conditions is realized by a unique Newton map. In an appendix, we introduce a class of bounded type transcendental entire functions with the property that its set of escaping points is organized in the form of unbounded rays. This fourth main result is joint work with Günter Rottenfußer, Lasse Rempe and Dierk Schleicher, and is part of an answer to a long-standing conjecture of Fatou and Eremenko.