On the geometry of unbounded wandering domains
Abstract
We study the geometry of unbounded wandering domains of entire functions using Arakelian approximation. First, we show that, given a uniformly accessible closed set contained in a strip, the connected components of its interior can be realized as escaping or oscillating wandering domains of some entire function. The iterates of the function are univalent on these wandering domains, and any unbounded wandering domain remains unbounded under iteration. Next, we construct escaping or oscillating wandering domains by lifting bounded wandering domains with prescribed geometry via the exponential map. In particular, there exists an entire function that has a half-plane as a wandering domain. Finally, we show that, in some precise sense, any simply connected open set can be approximated by escaping or oscillating wandering domains. As a direct consequence, we obtain wandering domains whose complements have arbitrarily small areas.
Related
- Essential spectral geometry of the Maxwell system in unbounded domains
- Absence of bounded-orbit wandering domains
- Bounded-orbit wandering domains do not exist
- Formalization of Sullivan's No Wandering Domains Theorem in Lean
- On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative
BibTeX
@misc{ucakar2026geometry,
title = {{On the geometry of unbounded wandering domains}},
author = {Beno Učakar},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.23842},
eprint = {2606.23842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.23842},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the geometry of unbounded wandering domains
AU - Beno Učakar
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2606.23842
UR - https://arxiv.org/abs/2606.23842
AB - We study the geometry of unbounded wandering domains of entire functions using Arakelian approximation. First, we show that, given a uniformly accessible closed set contained in a strip, the connected components of its interior can be realized as escaping or oscillating wandering domains of some entire function. The iterates of the function are univalent on these wandering domains, and any unbounded wandering domain remains unbounded under iteration. Next, we construct escaping or oscillating wandering domains by lifting bounded wandering domains with prescribed geometry via the exponential map. In particular, there exists an entire function that has a half-plane as a wandering domain. Finally, we show that, in some precise sense, any simply connected open set can be approximated by escaping or oscillating wandering domains. As a direct consequence, we obtain wandering domains whose complements have arbitrarily small areas.
ER -