An entire function with no fixed points and no invariant Baker domains

TL;DR

This paper constructs an entire function that lacks fixed points and invariant Baker domains, answering a question posed by Buff about the existence of such functions.

Contribution

It provides the first explicit example of an entire function with no fixed points and no invariant Baker domains, resolving a long-standing open problem.

Findings

Existence of an entire function with no fixed points.

Existence of an entire function with no invariant Baker domains.

Answers a question raised by Buff.

Abstract

We show that there exists an entire function which has neither fixed points nor invariant Baker domains. The question whether such a function exists was raised by Buff.