A complementary proof of Baker's theorem of completely invariant components for transcendental entire functions

TL;DR

This paper provides an alternative proof of Baker's theorem on the uniqueness of completely invariant Fatou components for transcendental entire functions, addressing a missing case in the original proof.

Contribution

It offers a new proof of Baker's theorem, filling a gap in the original argument and enhancing understanding of Fatou set invariance.

Findings

Confirmed at most one completely invariant Fatou component exists

Provided an alternative proof addressing the missing case

Strengthened the theoretical foundation of transcendental dynamics

Abstract

Baker proved that for transcendental entire functions there is at most one completely invariant component of the Fatou set. It was observed by Julien Duval that there is a missing case in Baker's proof. In this article we follow Baker's ideas and give some alternative arguments to establish the result.