Bounded-orbit wandering domains do not exist

Abstract

This paper establishes that transcendental entire functions do not have wandering Fatou components whose forward orbit is bounded. This settles a long-standing question in transcendental dynamics. A stronger statement is proved using the same method: the orbit of every point in a wandering domain of a transcendental meromorphic function is unbounded. In particular, transcendental meromorphic functions do not admit orbitally bounded wandering domains. The proofs rely mostly on uniform estimates for hyperbolic area, following the approach of Ye [arXiv:2609.23834v1].