We construct a transcendental entire function for which infinitely many Fatou components share the same boundary. This solves the long-standing open problem of whether Fatou components can be Lakes of Wada, and answers the analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. The Fatou components in our construction are wandering domains that escape to infinity. Our theorem also provides the first example of an entire function having a simply connected Fatou component whose closure has a disconnected complement, answering a recent question of Boc Thaler. Using the same techniques, we give new counterexamples to a conjecture of Eremenko concerning curves in the escaping set of an entire function. This is joint work with Lasse Rempe and James Waterman.
2021/10/15 金 15:00 - 17:00
University of Liverpool