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Multicritical pseudo-Siegel bounds and the optimality of the Brjuno condition for all neutral polynomials

  • Dzmitry Dudko
  • Yusheng Luo
  • Runze Zhang
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Abstract

We establish uniform multicritical pseudo-Siegel bounds for polynomials of Zakeri type, where all free critical points are on the boundary of an invariant Siegel disk of bounded type. This generalizes the unicritical result~\cite{DL:wZ}. As an application, we show the optimality of the Brjuno condition under the Zakeri restriction: as the rotation number becomes non-Brjuno, the Siegel disk evolves into the Cremer Hedgehog. Combined with~\cite{BCR26}, this implies the optimality of the Brjuno condition for all neutral polynomials, resolving a conjecture by Douady from the 1980s.

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Cite

BibTeXRIS
BibTeX
@misc{dudko2026multicritical,
  title = {{Multicritical pseudo-Siegel bounds and the optimality of the Brjuno condition for all neutral polynomials}},
  author = {Dzmitry Dudko and Yusheng Luo and Runze Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37468},
  eprint = {2610.37468},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37468},
}
RIS
TY  - UNPB
TI  - Multicritical pseudo-Siegel bounds and the optimality of the Brjuno condition for all neutral polynomials
AU  - Dzmitry Dudko
AU  - Yusheng Luo
AU  - Runze Zhang
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37468
UR  - https://arxiv.org/abs/2610.37468
AB  - We establish uniform multicritical pseudo-Siegel bounds for polynomials of Zakeri type, where all free critical points are on the boundary of an invariant Siegel disk of bounded type. This generalizes the unicritical result~\cite{DL:wZ}. As an application, we show the optimality of the Brjuno condition under the Zakeri restriction: as the rotation number becomes non-Brjuno, the Siegel disk evolves into the Cremer Hedgehog. Combined with~\cite{BCR26}, this implies the optimality of the Brjuno condition for all neutral polynomials, resolving a conjecture by Douady from the 1980s.
ER  -