Biggest bounded type Siegel disks of monic polynomials include those that stick to all critical points

Abstract

We prove that for all degree $d\geq 2$ and all bounded type irrational $θ$, in the space of monic polynomials having a period $1$ Siegel disk $Δ$ of rotation number $θ$, the maximum locus of the conformal radius of $Δ$ with respect to its fixed point contains polynomials having all critical points on the boundary of $Δ$. We apply this to reduce a conjecture of Douady (optimality of the Bruno condition) to a weaker statement.