Eremenko's conjecture, wandering Lakes of Wada, and maverick points

Abstract

We develop a general technique for realising full closed subsets of the complex plane as wandering sets of entire functions. Using this construction, we solve a number of open problems. (1) We construct a counterexample to Eremenko's conjecture, a central problem in transcendental dynamics that asks whether every connected component of the set of escaping points of a transcendental entire function is unbounded. (2) We prove that there is a transcendental entire function for which infinitely many Fatou components share the same boundary. This resolves the long-standing problem whether "Lakes of Wada continua" can arise in complex dynamics, and answers the analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. (3) We answer a question of Rippon concerning the existence of non-escaping points on the boundary of a bounded escaping wandering domain, that is, a wandering Fatou component contained in the escaping set. In fact we show that the set of such points can have positive Lebesgue measure. (4) We give the first example of an entire function having a simply connected Fatou component whose closure has a disconnected complement, answering a question of Boc Thaler. In view of (3), we introduce the concept of "maverick points": points on the boundary of a wandering domain whose accumulation behaviour differs from that of internal points. We prove that the set of such points has harmonic measure zero, but that it can nonetheless be rather large. For example, it may have positive planar Lebesgue measure.