Analytic centralisers of quadratic germs and the Brjuno dichotomy
Abstract
For Qα(z)=e2πiαz+z2 with α irrational, we prove that the analytic centraliser of Qα is trivial whenever Qα is not analytically linearizable. Together with the quadratic Brjuno--Yoccoz theorem this gives a complete dichotomy, for the centraliser is infinite cyclic at the non-Brjuno parameters and conjugate to the group of nonzero complex dilations at the Brjuno parameters. The result extends the dense-parameter conclusion of Avila--Cheraghi--Eliad to every nonlinearizable member of the quadratic family. The proof starts from Inou's graph-globalization construction, where we retain the original neutral fixed point on a Riemann surface and verify the branch topology and finite-valence estimate, including multiplicities. Nonlinearizability, through the equality case of Schwarz's lemma, forces the surface to be the plane, and a direct quadratic growth estimate then makes the two entire projections polynomial, so that the established polynomial commutant classification yields the integral-iterate conclusion. We also determine the convergence set of the formal commutants and classify the analytic iterative roots of every nonzero iterate.
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BibTeX
@misc{si2026analytic,
title = {{Analytic centralisers of quadratic germs and the Brjuno dichotomy}},
author = {Wen Si},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18667},
eprint = {2610.18667},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18667},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Analytic centralisers of quadratic germs and the Brjuno dichotomy
AU - Wen Si
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18667
UR - https://arxiv.org/abs/2610.18667
AB - For Q_α(z)=e^2πiαz+z^2 with α irrational, we prove that the analytic centraliser of Q_α is trivial whenever Q_α is not analytically linearizable. Together with the quadratic Brjuno--Yoccoz theorem this gives a complete dichotomy, for the centraliser is infinite cyclic at the non-Brjuno parameters and conjugate to the group of nonzero complex dilations at the Brjuno parameters. The result extends the dense-parameter conclusion of Avila--Cheraghi--Eliad to every nonlinearizable member of the quadratic family. The proof starts from Inou's graph-globalization construction, where we retain the original neutral fixed point on a Riemann surface and verify the branch topology and finite-valence estimate, including multiplicities. Nonlinearizability, through the equality case of Schwarz's lemma, forces the surface to be the plane, and a direct quadratic growth estimate then makes the two entire projections polynomial, so that the established polynomial commutant classification yields the integral-iterate conclusion. We also determine the convergence set of the formal commutants and classify the analytic iterative roots of every nonzero iterate.
ER -