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PreprintCC BY 4.0arXiv

Bounded-orbit wandering domains do not exist

  • Kostiantyn Drach
  • Leticia Pardo-Simón
  • Beno Učakar
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Abstract

This paper establishes that transcendental entire functions do not have wandering Fatou components whose forward orbit is bounded. This settles a long-standing question in transcendental dynamics. A stronger statement is proved using the same method: the orbit of every point in a wandering domain of a transcendental meromorphic function is unbounded. In particular, transcendental meromorphic functions do not admit orbitally bounded wandering domains. The proofs rely mostly on uniform estimates for hyperbolic area, following the approach of Ye [arXiv:2609.23834v1].

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Cite

BibTeXRIS
BibTeX
@misc{drach2026boundedorbit,
  title = {{Bounded-orbit wandering domains do not exist}},
  author = {Kostiantyn Drach and Leticia Pardo-Simón and Beno Učakar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30103},
  eprint = {2609.30103},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30103},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Bounded-orbit wandering domains do not exist
AU  - Kostiantyn Drach
AU  - Leticia Pardo-Simón
AU  - Beno Učakar
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30103
UR  - https://arxiv.org/abs/2609.30103
AB  - This paper establishes that transcendental entire functions do not have wandering Fatou components whose forward orbit is bounded. This settles a long-standing question in transcendental dynamics. A stronger statement is proved using the same method: the orbit of every point in a wandering domain of a transcendental meromorphic function is unbounded. In particular, transcendental meromorphic functions do not admit orbitally bounded wandering domains. The proofs rely mostly on uniform estimates for hyperbolic area, following the approach of Ye [arXiv:2609.23834v1].
ER  -