Singular orbits and Baker domains

TL;DR

This paper constructs a transcendental meromorphic function with a Baker domain where all singular values are super-attracting periodic points, answering longstanding questions and demonstrating the domain's large annuli can be arbitrarily chosen.

Contribution

It provides a novel example of a meromorphic function with a Baker domain containing only super-attracting singular values, resolving open questions in complex dynamics.

Findings

Existence of a meromorphic function with a Baker domain with only super-attracting singular values

The Baker domain can contain arbitrarily large annuli of definite modulus

Answers to questions posed by Bergweiler (1993) and Mihaljević and the author (2013)

Abstract

We show that there is a transcendental meromorphic function with an invariant Baker domain $U$ such that every singular value of $f$ is a super-attracting periodic point. This answers a question of Bergweiler from 1993. We also show that $U$ can be chosen to contain arbitrarily large round annuli, centred at zero, of definite modulus. This answers a question of Mihaljevi\'c and the author from 2013, and complements recent work of Bara\'nski et al concerning this question.