Rigidity of the attractor of neutral renormalization
Abstract
We prove combinatorial rigidity for the full renormalization attractor of neutral quadratic polynomials. More precisely, any two bi-infinite renormalization towers with the same combinatorics are conformally conjugate on neighborhoods of their Mother Hedgehogs. The rigidity theorem includes arbitrary irrational combinatorics and respective parabolic enrichments. The proof relies on a comprehensive analysis of neutral cascades, i.e. transcendental dynamical systems arising from the rescaled limits of the first return maps of neutral quadratic polynomials. We establish uniform butterfly bounds for neutral cascades and prove that any two combinatorially equivalent neutral cascades are affinely conjugate. We also prove rigidity for parabolic towers with equivalent backward combinatorics.
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BibTeX
@misc{dudko2026rigidity,
title = {{Rigidity of the attractor of neutral renormalization}},
author = {Dzmitry Dudko and Willie Rush Lim and Mikhail Lyubich},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37383},
eprint = {2609.37383},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37383},
}
RIS
TY - UNPB
TI - Rigidity of the attractor of neutral renormalization
AU - Dzmitry Dudko
AU - Willie Rush Lim
AU - Mikhail Lyubich
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37383
UR - https://arxiv.org/abs/2609.37383
AB - We prove combinatorial rigidity for the full renormalization attractor of neutral quadratic polynomials. More precisely, any two bi-infinite renormalization towers with the same combinatorics are conformally conjugate on neighborhoods of their Mother Hedgehogs. The rigidity theorem includes arbitrary irrational combinatorics and respective parabolic enrichments. The proof relies on a comprehensive analysis of neutral cascades, i.e. transcendental dynamical systems arising from the rescaled limits of the first return maps of neutral quadratic polynomials. We establish uniform butterfly bounds for neutral cascades and prove that any two combinatorially equivalent neutral cascades are affinely conjugate. We also prove rigidity for parabolic towers with equivalent backward combinatorics.
ER -