Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains

TL;DR

This paper extends the understanding of natural families of entire functions beyond finite-type cases, introduces a complex manifold structure for quasiconformally equivalent functions, and constructs a new multiplier-like object for wandering domains.

Contribution

It generalizes existing results to broader classes of entire functions and introduces a novel distortion sequence for wandering domains in complex dynamics.

Findings

The set of quasiconformally equivalent entire functions forms a complex manifold.

A new distortion sequence acts as a multiplier analogue for wandering domains.

Under certain conditions, the distortion sequence varies analytically within parameter families.

Abstract

Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of singular values has finite cardinality). Recent developments in the field go beyond this setting. In this paper we extend Eremenko and Lyubich's result on natural families of entire maps to the case where the set of singular values is not the entire complex plane, showing under this assumption that the set $M_f$ of entire functions quasiconformally equivalent to $f$ admits the structure of a complex manifold (of possibly infinite dimension). Moreover, we will consider functions with wandering domains -- another hot topic of research in complex dynamics. Given an entire function $f$ with a simply connected wandering domain $U$, we construct an analogue of the multiplier of a periodic orbit, called a distortion sequence, and show that, under some hypotheses, the distortion sequence moves analytically as $f$ moves within appropriate parameter families.