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PreprintCC BY 4.0arXiv

Hyperbolic area of Fatou components and new proofs of no-wandering-domain theorems

  • Zihao Ye
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Abstract

We give a new elementary proof of Sullivan's no-wandering-domain theorem for rational maps of degree at least two using hyperbolic area. We then develop this method further to prove that a transcendental entire function has no wandering Fatou components if its set of singular values is compact and all accumulation points of that set lie in the Fatou set. Neither proof uses quasiconformal techniques.

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Cite

BibTeXRIS
BibTeX
@misc{ye2026hyperbolic,
  title = {{Hyperbolic area of Fatou components and new proofs of no-wandering-domain theorems}},
  author = {Zihao Ye},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.23834},
  eprint = {2609.23834},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23834},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Hyperbolic area of Fatou components and new proofs of no-wandering-domain theorems
AU  - Zihao Ye
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.23834
UR  - https://arxiv.org/abs/2609.23834
AB  - We give a new elementary proof of Sullivan's no-wandering-domain theorem for rational maps of degree at least two using hyperbolic area. We then develop this method further to prove that a transcendental entire function has no wandering Fatou components if its set of singular values is compact and all accumulation points of that set lie in the Fatou set. Neither proof uses quasiconformal techniques.
ER  -