Analytic centralisers of quadratic germs and the Brjuno dichotomy

Abstract

For $Q_α(z)=e^{2πiα}z+z^2$ 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.