Slow escaping points of meromorphic functions

TL;DR

The paper proves that transcendental meromorphic functions have points in their Julia set that escape to infinity arbitrarily slowly, introduces various slow escaping sets, and explores their structure and relation to the Julia set.

Contribution

It establishes the existence of slow escaping points in the Julia set of meromorphic functions and introduces new sets characterizing different escape rates.

Findings

Existence of points in Julia set escaping arbitrarily slowly.

Introduction of several slow escaping sets with bounded escape rates.

Connections between slow escaping sets and Julia set structure.

Abstract

We show that for any transcendental meromorphic function $f$ there is a point $z$ in the Julia set of $f$ such that the iterates $f^n(z)$ escape, that is, tend to $\infty$, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which $f^n(z)$ tends to $\infty$ at a bounded rate, and establish the connections between these sets and the Julia set of $f$. To do this, we show that the iterates of $f$ satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a plane-filling wandering domain. We give examples to show how varied the structures of these slow escaping sets can be.