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

Abstract

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

22 pages 2 figures