Baker domains and orbits disappearing to infinity

TL;DR

This paper investigates the dynamics of transcendental entire functions, revealing how attracting orbits escaping to infinity relate to Baker domains and how functions with such domains can be approximated by those with attracting fixed points.

Contribution

It establishes a connection between attracting fixed points with multipliers tending to one and the existence of doubly parabolic Baker domains, and shows approximation results within quasiconformal classes.

Findings

Attracting fixed points escaping to infinity with multipliers tending to one lead to doubly parabolic Baker domains.

Functions with doubly parabolic Baker domains can be approximated by functions with attracting fixed points.

The study provides a characterization of the boundary behavior of certain transcendental entire functions.

Abstract

We study attracting orbits escaping to infinity in natural families of transcendental entire functions. We show that, if an attracting fixed point escapes to infinity while its multiplier tends to one, then the limiting function has a doubly parabolic Baker domain. Conversely, we show that any function with an invariant doubly parabolic Baker domain can be approximated locally uniformly by functions in its quasiconformal equivalence class having an attracting fixed point whose multiplier tends to one.