Linearizability of Saturated Polynomials

Abstract

Brjuno and Rüssmann proved that every irrationally indifferent fixed point of an analytic function with a Brjuno rotation number is linearizable, and Yoccoz proved that this is sharp for quadratic polynomials. Douady conjectured that this is sharp for all rational functions of degree at least 2, i.e., that non-Möbius rational functions cannot have Siegel disks with non-Brjuno rotation numbers. We prove that Douady's conjecture holds for the class of polynomials for which the number of infinite tails of critical orbits in the Julia set equals the number of irrationally indifferent cycles. As a corollary, Douady's conjecture holds for the polynomials $P(z) = z^d + c$ for all $d > 1$ and all complex $c$.