Multicritical pseudo-Siegel bounds and the optimality of the Brjuno condition for all neutral polynomials

Abstract

We establish uniform multicritical pseudo-Siegel bounds for polynomials of Zakeri type, where all free critical points are on the boundary of an invariant Siegel disk of bounded type. This generalizes the unicritical result~\cite{DL:wZ}. As an application, we show the optimality of the Brjuno condition under the Zakeri restriction: as the rotation number becomes non-Brjuno, the Siegel disk evolves into the Cremer Hedgehog. Combined with~\cite{BCR26}, this implies the optimality of the Brjuno condition for all neutral polynomials, resolving a conjecture by Douady from the 1980s.