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Lebesgue measure of the postcritical set of neutral quadratic polynomials

  • Willie Rush Lim
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Abstract

We prove that the postcritical set of every quadratic polynomial with a neutral fixed point has zero Lebesgue measure. In particular, no arithmetic condition on the rotation number is required. Our proof uses sector renormalization and the pseudo-Siegel bounds of Dudko-Lyubich to derive uniform distortion estimates for the renormalization change of variables. For sector renormalization towers, we also prove the same zero area statement for the associated postcritical set and the optimal Brjuno criterion for the interior of the associated Mother Hedgehog.

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Cite

BibTeXRIS
BibTeX
@misc{lim2026lebesgue,
  title = {{Lebesgue measure of the postcritical set of neutral quadratic polynomials}},
  author = {Willie Rush Lim},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35563},
  eprint = {2609.35563},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35563},
}
RIS
TY  - UNPB
TI  - Lebesgue measure of the postcritical set of neutral quadratic polynomials
AU  - Willie Rush Lim
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35563
UR  - https://arxiv.org/abs/2609.35563
AB  - We prove that the postcritical set of every quadratic polynomial with a neutral fixed point has zero Lebesgue measure. In particular, no arithmetic condition on the rotation number is required. Our proof uses sector renormalization and the pseudo-Siegel bounds of Dudko-Lyubich to derive uniform distortion estimates for the renormalization change of variables. For sector renormalization towers, we also prove the same zero area statement for the associated postcritical set and the optimal Brjuno criterion for the interior of the associated Mother Hedgehog.
ER  -