Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains
Author(s)
Ferreira, Gustavo
van Strien, Sebastian
Type
Journal Article
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.
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.
Date Issued
2025-05-01
Date Acceptance
2025-01-24
Citation
Mathematische Annalen, 2025, 392 (1), pp.701-732
ISSN
0025-5831
Publisher
Springer
Start Page
701
End Page
732
Journal / Book Title
Mathematische Annalen
Volume
392
Issue
1
Copyright Statement
© The Author(s) 2025 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Identifier
10.1007/s00208-025-03107-8
Subjects
Mathematics Subject Classification 37F10
37F44
37F46
Publication Status
Published
Date Publish Online
2025-02-15