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PreprintCC BY 4.0arXiv

Uniformization of Julia sets of geometrically finite rational maps

  • Yusheng Luo
  • Dimitrios Ntalampekos
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Abstract

A Schottky set is the complement of a collection of disjoint open disks in the Riemann sphere. We completely characterize geometrically finite rational maps whose Julia sets are quasiconformally equivalent to Schottky sets. We also prove that the closure of the basin of infinity of a geometrically finite unicritical polynomial is quasiconformally equivalent to a Schottky set if and only if either there exists an attracting cycle and all bounded Fatou components have disjoint closures or there exists a parabolic cycle of multiplicity three. An ingredient of the proof is a new dynamical characterization of pairs of arbitrary quasicircles that are quasiconformally equivalent to pairs of tangent circles.

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Cite

BibTeXRIS
BibTeX
@misc{luo2026uniformization,
  title = {{Uniformization of Julia sets of geometrically finite rational maps}},
  author = {Yusheng Luo and Dimitrios Ntalampekos},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21945},
  eprint = {2610.21945},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21945},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Uniformization of Julia sets of geometrically finite rational maps
AU  - Yusheng Luo
AU  - Dimitrios Ntalampekos
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21945
UR  - https://arxiv.org/abs/2610.21945
AB  - A Schottky set is the complement of a collection of disjoint open disks in the Riemann sphere. We completely characterize geometrically finite rational maps whose Julia sets are quasiconformally equivalent to Schottky sets. We also prove that the closure of the basin of infinity of a geometrically finite unicritical polynomial is quasiconformally equivalent to a Schottky set if and only if either there exists an attracting cycle and all bounded Fatou components have disjoint closures or there exists a parabolic cycle of multiplicity three. An ingredient of the proof is a new dynamical characterization of pairs of arbitrary quasicircles that are quasiconformally equivalent to pairs of tangent circles.
ER  -