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Biggest bounded type Siegel disks of monic polynomials include those that stick to all critical points

  • Xavier Buff
  • Arnaud Chéritat
  • Pascale Roesch
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Abstract

We prove that for all degree d≥ 2 and all bounded type irrational θ, in the space of monic polynomials having a period $1$ Siegel disk Δ of rotation number θ, the maximum locus of the conformal radius of Δ with respect to its fixed point contains polynomials having all critical points on the boundary of Δ. We apply this to reduce a conjecture of Douady (optimality of the Bruno condition) to a weaker statement.

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Cite

BibTeXRIS
BibTeX
@misc{buff2026biggest,
  title = {{Biggest bounded type Siegel disks of monic polynomials include those that stick to all critical points}},
  author = {Xavier Buff and Arnaud Chéritat and Pascale Roesch},
  howpublished = {HAL},
  year = {2026},
  month = jun,
  doi = {10.48550/arxiv.2606.24833},
  eprint = {2606.24833},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05738283v1},
}
RIS
TY  - UNPB
TI  - Biggest bounded type Siegel disks of monic polynomials include those that stick to all critical points
AU  - Xavier Buff
AU  - Arnaud Chéritat
AU  - Pascale Roesch
PY  - 2026
DA  - 2026/06/23
PB  - HAL
DO  - 10.48550/arxiv.2606.24833
UR  - https://hal.science/hal-05738283v1
AB  - We prove that for all degree d≥ 2 and all bounded type irrational θ, in the space of monic polynomials having a period $1$ Siegel disk Δ of rotation number θ, the maximum locus of the conformal radius of Δ with respect to its fixed point contains polynomials having all critical points on the boundary of Δ. We apply this to reduce a conjecture of Douady (optimality of the Bruno condition) to a weaker statement.
ER  -