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Uniform bounds for bubbles of neutral quadratic polynomials
Abstract
Given a quadratic polynomial with an irrationally indifferent fixed point of bounded type, a bubble is an iterated preimage of its Siegel disk. Similar to pseudo-Siegel disks, one constructs pseudo-bubbles from bubbles by filling in parabolic fjords. We introduce the near-degenerate regime for the α'-Dynamics controlling the geometry of these pseudo-bubbles in deep scale. Consequently, we obtain uniform bounds on the size and regularity of pseudo-bubbles independent of the rotation number. In [DLL], jointly with Lyubich, this result serves as a necessary ingredient for establishing uniform butterfly bounds and ultimately the combinatorial rigidity of the full attractor of neutral renormalization.
Related
- Lebesgue measure of the postcritical set of neutral quadratic polynomials
- Multicritical pseudo-Siegel bounds and the optimality of the Brjuno condition for all neutral polynomials
- Iterated Monodromy Groups of Quadratic Polynomials, I
- Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials
- Uniform comparison of multiplier and test ideals for polynomials with affinely independent support
BibTeX
@misc{dudko2026uniform,
title = {{Uniform bounds for bubbles of neutral quadratic polynomials}},
author = {Dzmitry Dudko and Willie Rush Lim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15500},
eprint = {2609.15500},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15500},
}
RIS
TY - UNPB
TI - Uniform bounds for bubbles of neutral quadratic polynomials
AU - Dzmitry Dudko
AU - Willie Rush Lim
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15500
UR - https://arxiv.org/abs/2609.15500
AB - Given a quadratic polynomial with an irrationally indifferent fixed point of bounded type, a bubble is an iterated preimage of its Siegel disk. Similar to pseudo-Siegel disks, one constructs pseudo-bubbles from bubbles by filling in parabolic fjords. We introduce the near-degenerate regime for the α'-Dynamics controlling the geometry of these pseudo-bubbles in deep scale. Consequently, we obtain uniform bounds on the size and regularity of pseudo-bubbles independent of the rotation number. In [DLL], jointly with Lyubich, this result serves as a necessary ingredient for establishing uniform butterfly bounds and ultimately the combinatorial rigidity of the full attractor of neutral renormalization.
ER -